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«  '  0 


i 


BY  Tin*]  ,SAMK  AUTIIUR. 


ALGEBRA    FOR    COLLEGES. 

KEY    TO   ABOVE. 

ELEMENTS    OP    GEOMETRY. 

ELEMENTS  OP  PLANE  AND  SPHER- 
ICAL   TRIGONOMETRY. 


ASTRONOMY.      For  Students  and  Goncnil 
Readers.     By  Simon  Kkwcomu  and  En- 

AVAUD  S.    IIOI.DEN. 


I.  iJ . 


1     1 


NM  W  COMB'S  MA  2  U  EM  A  TWA  L    SERIES. 


ALGEBKA 


FOR 


SCHOOLS    AND    COLLEGES 


BY 


I.. 


\'. 


:v 


SIMON    NEWOOMB    ^- . .  ;;t;  ;..;_^.- 

Profesnor  of  3fa(hematk.%   United  States  Nai'y 
TIIIRD   EDITION.  REVISED. 


y 


NEW  YORK 
HENRY   HOLT  AND  COMPANY 

1882 


V*   \ 


V 


COPYUtGHT,   1881, 
BY 

Henry  Hoi/r  &  Co, 


^J 


ELF.CTRnTVrED    BY 

SMITH  &  McDOtlGAL, 
82  Deekman  St.,  N.  Y, 


\ 


P  R  E  F  x\.  C  E . 


The  course  of  algebra  embodied  in  the  present  work  is  substantially 
that  pursued  by  students  in  our  besi  preparatory  and  scientific  schools 
and  colleges,  with  such  extensions  as  seemed  necessary  to  afford  an 
improved  basis  fur  more  advanced  studies.  For  the  convenience  of 
teachers  the  work  is  divided  into  two  parts,  the  first  adapted  to  well- 
prepared  beginners  and  comprising  about  what  is  conunonly  required  for 
admission  to  college ;  and  the  second  designed  for  the  more  advanced 
general  student,-  As  the  w^rk  deviates  in  several  points  from  the  models 
most  familiar  to  our  teachers,  a  statement  of  the  principles  on  which  it  is 
constructed  may  be  deemed  appropriate. 

One  well-known  principle  underlying  the  acquisition  of  knowledge  is 
that  an  idea  cannot  be  fully  grasped  by  the  youthful  mind  unless  it  is 
presented  under  a  concrete  form.  Whenever  possible  an  abstrart  idea 
nmst  be  embodied  in  some  visible  representation,  and  all  general  theorems 
must  be  presented  in  a  variety  of  special  forms  in  which  they  may  be 
seen  inductively.  In  accordance  with  this  principle,  numerical  exam- 
ples of  nearly  all  algebraic  operations  and  theorems  have  been  presented. 
For  the  purpose  of  illustrauon,  numbers  have  been  preferred  to  literal 
synabols  when  they  would  serve  the  purpose  equally  well.  The  relations 
of  positive  and  negative  algebraic  quantities  have  been  represented  by 
lines  and  directions  from  tlie  beginning  in  order  that  the  pupil  might  be 
able  to  give,  not  only  a  numerical,  but  a  visible,  meaning  to  all  algebraic 
quantities.  Should  it  appear  to  any  one  that  we  thus  detract  from  the 
generality  of  algebraic  quantities,  it  is  sufficient  to  reply  that  the  system 
is  thiC  same  which  mathematicians  use  to  assist  their  conceptions  of 
advanced  algebra,  and  without  which  they  would  never  have  been  able 
to  grasp  the  complicated  relations  of  imaginary  (juantities.     Algebraic 


IV 


PREFACE. 


operations  with  pure  numbers  ure  nuido  to  precede  the  use  oi"  aymbols, 
;iii(l  tii(!  latter  are  introdnc(!(l  only  aCtor  tiir  pupil  has  had  a  certain 
amount  of  familiarity  with  the  distinction  between  algebraic  and  numer- 
ical opcrationn. 

Another,  but,  unfortunately,  a  less  familiar  fact  is,  that  all  mathematical 
conceptions  require  time  to  become  engrafted  upon  the  mind,  and  the 
more  time  the  greater  their  abstruseness.  It  is,  the  author  conceives, 
from  a  failure  to  take  account  of  this  fact,  rather  than  from  any  inherent 
defect  in  the  minds  of  our  youth,  that  we  are  to  attribute  the  backward 
state  of  mathematical  instruction  in  this  country,  as  compared  \vith  the 
continent  of  Europe.  Let  us  take  for  instance  the  case  of  the  student 
commencing  the  calculus.  On  the  system  which  was  almost  universal 
among  us  a  few  years  ago,  and  which  is  still  widely  prevalent,  he  is  con- 
fronted at  the  outset  with  a  number  of  entirely  new  conceptions,  such 
as  those  of  variables,  functions,  increments,  infinitesimals  and  limits. 
In  his  first  lesson  he  finds  these  all  combined  with  a  notation  so  entirely 
different  from  that  to  which  he  has  been  accustomed,  that  before  the 
new  ideas  and  forms  of  thought  can  take  permanent  root  in  his  mind, 
he  is  through  with  the  subject,  and  all  that  he  hsis  learned  is  apt  to  vanish 
from  his  memory  in  a  few  months. 

The  author  conceives  that  the  true  method  of  meeting  this  difficulty  is 
to  adopt  the  French  and  Grcrman  plan  of  teaching  algebra  in  a  broader 
way,  and  of  introducing  the  more  advanced  conceptions  at  the  earliest 
practicable  period  in  the  course  Accordingly,  the  attempt  is  made  in  the 
present  work  to  introduce  each  advanced  conception,  disguised  perhaps 
under  some  simple  form,  in  advance  of  its  general  enunciation  and  at  as 
early  a  period  as  the  student  can  be  expected  to  understand  it.  In  doing 
this,  logical  order  is  frequently  sacrificed  lu  the  exigencies  of  the  case, 
because  there  are  several  subjects  with  which  a  certain  amount  of  famil- 
iarity must  be  acquired  before  the  pupil  can  even  clearly  comprehend 
general  statements  respecting  them. 

A  Aird  feature  of  the  work  is  that  of  subdividing  each  subject  as 
minutely  as  possible,  and  exercising  the  pupil  on  the  details  preparatory  to 
combining  them  into  a  whole.  To  cite  one  or  two  instances :  a  difficulty 
which  not  only  the  beginner  but  the  expert  mathematician  frequently 
meets  is  that  of  stating  his  conceptions  in  algebraic  language.  Exercises 
in  such  statements  have  therefore  been  made  to  precede  any  solution  of 


i 


f 


i 


1 


L 


Plih'FACE.  V 

problems.     In  peiiprnl  each  principle  which  i,s  to  he  prPHentod  or  ut*v(\  is 
Htiitcd  Hiiij?ly,  and  the  pupil  irt  practiced  u|»'tn  it  before  proceeding  to 

uiiotlicr. 

riulijccts  have  tor  the  most  part  been  omitted  which  do  not  find  aj)pli- 
cation  either  in  the  n\  k  itself  or  in  .subaeciuent  partH  of  the  usual  course 
o?  mathematics,  or  whic>i  do  not  conduce  to  a  uiutheniatical  t'-aining. 
Sturm's  Theorem  has  been  entirely  omitted,  and  a  more  sinipk^  process 
substituted.  The  subject  of  the  greatest  common  divisor  of  two  polyno- 
mials has  been  postponed  to  what  the  author  considers  its  proper  place, 
in  the  general  theory  of  equations.  It  has,  however,  been  presentv^d  in 
such  a  form  that  it  can  be  taught  to  puiuls  preparing  for  colleges  where 
it  is  stili  re(piired  for  admission. 

Thoroughness  at  each  step  has  been  aimed  at  rather  than  multiplicity 
of  subjects.  It  is,  the  author  :'onc(nves^  a  gi'eat  ami  too  common 
mi.^take  to  present  a  matheniatioal  subject  to  the  mind  of  tlie  student 
wtrkout  sufTicient  f.iliioss  of  explanation  a.nd  variety  of  illustration  to 
enable  hini  to  comprelicnd  and  iip|i!y  it.  II'  lie  has  not  time  to  master  a 
complete  course,  it  is  bette"  to  omit  entirely  what  is  least  necessary  th.an 
to  gain  time  by  going  rapidly  over  a  great  number  of  things.  Some 
hints  to  those  who  may  not  liave  time  to  master  the  whole  work  may 
there.' )'v  be  acceptable. 

Pa.t  T  is  essential  to  every  one  desiring  to  make  use  of  algebra.  Book 
VIII,  especially  tlic  concluding  sections  on  notation,  is  to  be  thoroughly 
mastered,  before  going  farther,  as  forming  tlie  foundation  of  advanced 
alf^ebra ;  and  affording  n  very  easy  and  valuable  disciplire  in  the  language 
vl'  mathematics.  Afterward,  a  selection  may  bo  made  according  to  cir- 
cumstances. The  student  who  is  pursuing  the  subject  for  the  sole 
purpose  of  liberal  training,  and  without  intending  to  advance  beyoml  it, 
will  find  the  theories  of  numbers  and  the  combinatory  analysis  most, 
worthy  of  stud^ .  The  theory  of  probabilities  and  the  method  in  which 
it  is  applied  to  such  practical  questions  as  those  connected  with  insurance 
will  be  of  especial  value  in  training  his  judgment  to  the  affairs  of  life. 

The  student  who  intentls  to  take  a  full  course  of  mathematics  with  a 
view  of  its  application  to  physics,  engineering,  or  other  subjects,  may,  if 
necessary,  omit  the  book  on  the  theoiy  of  numbers,  and  portions  of  tlu' 
chapter  on  the  snmmatiou  of  series.  Fiuiciions  and  the  fiuictiiinal  notation, 
the  doctrine  of  limits,  and  the  general  theory  of  equaliour  will  claim  his 


V'l 


pr/:fa('a\ 


especial  attention,  while  the  theory  of  iiiaginary  qnnntitirs  will  be  studied 
iiiJiiiily  to  secure  thoroufjhncss  in  subsequent  parts  of  his  course. 

As  it  lifus  frequently  been  u  part  of  tlie  author's  duty  to  ascertain  what 
is  really  left  of  a  ;;uurse  of  matliematical  study  in  the  minds  of  those 
who  have  been  through  college,  some  hints  on  the  best  methods  of 
study  in  conncjction  with  the  present  work  may  bo  excused.  If  askeil 
to  point  out  the  greatest  error  in  our  usual  system  of  mathematical 
instruction  from  the  conunon  school  upward,  he  would  reply  that  it  con- 
sisted in  expending  too  much  of  the  mental  power  of  the  student  upon 
j)roblem3  and  exeicises  above  Iiis  capacity.  With  tiie  exception  of  tho 
lundamental  routine-operations,  problems  and  exercises  should  be  confined 
to  insuring  a  proper  understanding  of  the  principles  involved:  this  onco 
ascertained,  it  is  better  that  the  student  should  go  on  rather  than  expend 
time  in  doing  what  it  is  certain  he  can  do.  Problems  of  some  difficuity 
are  found  among  the  exercises  of  the  present  work;  they  are  inserted 
rather  to  give  the  teacher  a  good  choice  from  which  to  select  than  to 
require  that  any  student  should  <lo  them  all. 

It  would,  the  author  conceives,  be  found  an  improvement  on  our  usual 
system  of  teaching  algebra  and  geometry  successively  if  the  analytic  and 
the  geometric  courses  of  mathematics  were  pursued  simultaneously.  The 
former  would  include  algebra  and  the  calculus,  the  latter  elementary 
geonietiy,  trigonometry,  and  analytic  geometry.  The  analytic  course 
woulil  then  furnish  nuithods  for  the  geometric  one,  and  tlie  latter  would 
furnisli  ;ipi)lications  and  illustrations  for  the  analytic  one. 

The  Key  to  the  work,  which  will  be  issued  as  soon  as  practicable,  will 
contain  not  only  the  usual  solutions,  but.  tho  explanations  and  demonstra- 
tions of  the  less  familiar  theorems,  ami  a  number  of  additional  problems. 

The  author  desires,  in  conclusion,  to  express  his  obligation  tu  the  many 
frienils  who  have  given  him  suggestions  respecting  the  work,  and  espe- 
cially to  Professor  J.  Howard  Gore  of  the  Columbian  University  who 
has  furnished  solutions  to  most  of  the  problems,  and  given  the  benefit  of 
his  experience  on  many  points  of  detail. 


TAJ3LE    OF    COXTEXTS. 


PAUT    1. -ELEMENTAKY    COURSE. 

BOOK   I.— THE   AL(JEDUAIC   LANCJUAGE. 

riiArTLu  I. — Algeuuaic  Numueks  and  Operations,  3.  Otncral 
Definitions,  3.  Algebraic  Numburs,  4.  Algebraic  Additiuu,  G, 
Subtraction,  8.     Multiplication,  9.     Division,  11. 

CiiAPTEu  II. — Algebraic  Sym  ;)ls,  Vi.  SyinlM)ls  of  Quantity,  12. 
Signs  of  Operation,  13. 

Chapter  III.— Formation  of  Compound  Expressions,   17.      Funda 
mental  Principles,  17.     Definitions.  1!>. 

CuAPTER  IV.— Construction  of  Algebraic  Expressions,  22.  Exer- 
cises in  Algebraic  Language,  25. 

BOOK    II.— ALliEBKAlC  OPERATIONS. 


General  Remarks,  28.     Definitions,  28. 

Chapter  I. — Algebraic  Addition  and  Subtraction,  30.  Algebraic 
Addition,  30.  Algebraic  Subtraction,  33.  Clearing  of  Parenthe- 
ses, 35.     Compound  Parentheses,  37. 

Cilvpter  II. — Multiplication,  38.  General  Laws  of  Multiplication,  38. 
Multiplication  of  Positive  Monomials,  40.  Rule  of  Signs  in 
Multiplication,  41.  Products  of  Polynomials  by  Monomials,  44. 
Multiplication  of  Polynomials  by  Polynomials,  47. 

Chapter  III. — Division,  52.  Division  of  Monomials  by  Monomials,  52. 
Rule  of  Signs  in  Division,  53.     Division  of  Polynomials  by  Mono- 


minis,  .ll.  ^"'artors  nnd  Multiplrs.  55.  FurtnrH  of  RlnoniinlH.  58. 
li)HHt  ('oininoii  Multipli'.  01  Division  of  ono  PoiviKiiuia!  Iiy 
aiiotiuT,  (W. 

CiiAi'TKU  IV.— Ok  Alokbraic  Fh actionh,  (57.  No>?ntivp  EsponcntH,  Tl. 
Dissection  of  FructionH,  73.  Atf>,'rc^'ati(in  of  Fractions,  74.  Factor- 
in)?  Fractions,  78.  Multiplication  and  Division  of  Fractions,  71). 
Division  of  one  Fraction  by  another,  82.  Reciprocal  ilelations  of 
Multiplication  and  Division,  H;{. 


BOOK    111.— OF  EQUATIONS. 

CllAPTRR  I— TiiK  Hediction  OF  Ecjl'ATiONs,  85.  AxioniH,  87.  Opera- 
tions of  Addition  and  Subtraction — transposinfr  Tenns,  87. 
Operation  of  Multiplication.  Si).  Heductlon  to  the  Normal  Form, 
5)0.     Degree  of  E<| nations,  Wo. 

ClIAI'TER  II. — E(iU.VTIONS  OF  TIM',  FlHST  DlXiUEK  WITH  ONK  UN- 
KNOWN QfANTlTY,  1)4.  Froldenis  leading  to  Simple  EcjuatiouH,  !)l). 
Problem  of  the  Couriers,  105.     Problems  of  Circular  Motion,  108. 

Chapter  III.— EiiUATiONs  of  the  First  Deghek  with  Sevkral 
Unknown  Qiantities,  109.  Equations  with  Two  Unknown 
Quantiti(^s,  109.  Solution  of  a  pair  of  Simultaneous  Equations 
containing  Two  Unknown  Quantities,  109.  Elimination  by  Com- 
parison, 110.  Elimination  by  Substitution,  111.  Elimination  by 
Addition  or  Subtraction,  112.  Problem  of  the  Sum  and  Differ- 
ences, 113.  Eiiuation.s  of  the  First  Degree  with  Three  or  More 
Unkno\i'n  Quantities,  116.  Elimination,  110.  Equivalent  and 
Inconsistent  Ecjuations,  131. 

Ciiai'ter  IV. — Of  Inequalities,  123. 


BOOK   IV.— RATIO   AND  PROPORTION. 
Chapter  I. — NATimE  of  a  Ratid,  128.    Properties  of  Ratios,  132. 

Chapter  II. — Proportion,  13;3.     Tlu'orems  of  Proportion,  134.    The 
Mean  Proportional,  lUS.     Multiple  Proportions,  13'J. 


CONTHSrs 


IX 


BOOK    V— OF    POWKHS   AND    HOOTS. 

Chapter  I. — Invomition,  lit.  liivoluticu  «»t'  I'nxliK^trt  and  CiuotipiitH, 
144.  Involution  of  I'owcrH,  Itr).  ('uhc  of  Nr>,-utivu  ExponriitH,  147. 
Aljj^cbniic  Sij^ii  of  Powith,  14b.  Involution  of  Hinoniialri — tlio 
Minoniiul  Tlicort'in,  148.     yquaru  of  u  Polynomial,  l.W. 

('II.M'TKII   II. — FiVOLUTION   AND   FllAniONAI,   EXPONENTS,   155.      Powoffl 

of  Exprc'HaionH  with  Fractional  Exixjnuntti,  157. 

CiiAPTEU  III.— Rkimction  OK  IiMiAnoNAi,  ExiMUCfsiONS,  l.'JO.  ncflni- 
tions,  \7}\\.  A^fjj;reji:ution  of  Similar  Torms,  KiO.  Factoring  Snrtls, 
KM.  Pi-rft'ct  iiqmuvy,  10(5  To  ('omi)leto  tho  Square,  107.  Irra- 
tioiuil  Factors,  1(>9. 


iVKllAL 

nUnown 

nations 

•  Com- 

ion  by 

Ditt'cr- 

or  More 

lit  uud 


32. 

4.     Tlic 


f 


BOOK  VI.— EQUATIONS  UEQUIHING  IRRATIONAL  OPERATIONS. 

CiiAJ'TEU  I.— Equations  with  Two  Tku.ms  only,  170.  Sf»lution  of  a 
liinomial  Equation,  170.  Special  Forms  of  Binomial  E(|uations, 
171.     Positive  and  Negative  Roots,  172. 

CnAPTEU  II. — Quadkatic  Equations,  174.  Solution  of  a  Complete 
Quadratic  Ecpiation,  17o.  Etjuations  which  may  he  reduced  to 
Quadratics,  179.  Factoring  a  Quadratic  Equation,  184.  Equations 
having  Imaginary  Roots,  188. 

Chaptek  III. — Reduction  op  Irrational  Equations  to  the  Normal 
Form,  189.     Clearing  of  Surds,  189. 

Chapter  IV. — Simultaneous  Quadratic  Equations,  193. 


BOOK   Vll— PROGRESSIONS. 

Chapter  I. — Arithmetical  Progression,  200.  Problems  in  Pro- 
gression, 202. 

Chapter  II.— Geometrical  Progression,  207.  Problems  of  Geo- 
metrical  Progression,  208.  Limit  of  the  Sum  of  a  Progression, 
211.    Comp<mnd  Interest,  217. 


X 


CONTEXTS. 


PART    II.— ADVANCED    COURSE. 

BOOK  VIII.— RELATIONS   BETWEEN   ALGEBRAJC  QUANTITIES. 

Function-s  and  their  Notation,  231.  Equations  of  the  First  Degree 
b«'twc'en  Two  Variables,  324.  Notation  of  Functions,  230.  Func- 
tions of  Sf'-eral  Variables,  232.  Use  of  Indices,  233.  Miscellaneous 
Function.s  of  Numbers,  235. 

BOOK   IX.— THE   THEORY   OF   NUMBERS. 

Chapter  I.— Tue  Divisiijii.ity  op  Numbers,  238.  Division  into  Prime 
Factors,  239.  Common  Divisors  of  two  numbers,  240.  Relations 
of  nu'nlx.Ts  to  their  Digits,  245.  Divisibility  of  Numbers  and  their 
Digits,  245.  Prime  Facttjrs  of  Numbers,  248.  Elementary  Theorems, 
251.     Binomial  Coefficients,  251.     Divisors  of  a  Number,  254. 

Chapter  II. — Of  Coxtinited  Fractions,  258.  Relations  of  Conirerging 
Fraction.',  267.     Periodic  Continued  Fractions,  270. 

BOOK   X.— THE  COMBINATORY   ANALYSIS. 

Chapter  I. — Permutations,  273.     Permutation  of  Sets,  275.    Circular 
,  Permutations,   277.     Permutations   when  Several  of  the  Things 
are  Identical,  279.     The  two  classes  of  Permutations,  281.     Sym- 
metric Functions,  284. 

Chapter  II. — Combinations,  285.  Combinations  with  'Repetition, 
287,  Sjiecial  Cases  of  Combinations,  289.  The  Binomial  Theorem 
when  the  Power  is  a  Whole  Number,  296. 

Chapter  III.— Theory  of  Probabilities,  299.    Probabilities  depend 
ing  up^m  Combinations,  300.     Compound  Events,  305.     Cases  of 
Unequal   Probability,  310.     Ai^plication  to  Life  Insurance,  316, 
Table  of  Mortalitv.  318. 


BOOK  XI.— OF  SERIES  AND  THE   DOCTRINE  OF  LIMITS. 
Chapter  I— Nature  of  a  Series,  331.    Notation  of  Sums,  324 


CONTENTS. 


XI 


■:  QUANTITIES. 

the  First  Degree 
tions,230.  Func- 
;;3.    Miscellaneous 

5EUS. 

)ivision  into  Prime 
rs,  240.  Relations 
Nural)er8  and  tlieir 
mentavy  Theorems, 
Number,  2f)4. 

tions  of  Con  (merging 
270. 

^LYSIS. 

Sets,  275.  Circular 
eral  of  the  Things 
itations,  281.     Sym- 


CuAPTER  II.— Development  in  Powers  of  a  Variable,  3£0. 
Method  of  Indeterminate  Coetlicieuts,  327.  Multiplicatioa  of  Two 
Infinite  Series,  333. 

Chapter  III.— Summation  op  Ser  s.  Of  Figurate  Numbers,  336. 
Enumeration  of  Triaufrulur  Piles  of  Shot,  339.  Sum  of  tlie 
Similar  Powers  of  an  Arithmcstical  Progression,  341,  Other  Series, 
346.     Of  Ditierences,  350.     Theorems  of  Diftereuces,  355. 

Chapter  IV.— The  Uoctuine  op  Limits,  358.  Notation  of  tho 
Method  of  Limits,  301.     Properties  of  Limits,  304. 

Chapter  V. — The  Binomial  and  Exponential  Theorems.  The 
Binomial  Theorem  for  all  values  of  the  Exponent,  308.  The 
Exponential  Theorem,  373. 

Chapter  VI. — Logarithms,  378.  Properties  of  Logarithms,  378.  Com- 
parison of  Two  Systems  of  Logarithm^;,  384. 

BOOK   Xn.— IMAGINARY  QUANTITIES. 

Chapter  I. — Operations  with  the  Imaginary  Unit,  391.  Addi- 
tion of  Imaginary  Expressions,  303.  Multi[)lication  of  Imaginary 
Quantities,  393.  Reduction  of  Functions  of  i  to  the  Normal 
Form,  390. 

Chapter  II.— The  Geometrical  Representation  of  Imaginary 
Quantities,  404. 


IS    with    lepetition, 
e  Binomial  Theorem 


robalnlities  depend- 
rcnts.  305.  Cases  of 
Life  Insurance,  310. 


[NE  OF  LIMITS, 
n  of  Sums,  324 


BOOK  XIII.— THE  GENERAL  THEORY  OF  EQUATIONS. 

Every  Ecjuation  has  a  Root,  416.  Number  of  Roots  of  General 
Equation,  418.  Relations  between  Coefficients  and  Roots,  425. 
Derived  Functions,  427.  Significance  of  the  Derived  Function,  430. 
Forms  of  the  Roots  of  E(juation,  431.  Decomi>osition  of  Rational 
Fractions,  433.  Greatest  Common  Divisor  of  Two  Functions,  438. 
Transformation  of  Equations,  4-42.  Resolution  of  Numerical  Equa- 
tions, 447. 


1 


FIRST  PART. 


ELEMENTARY   COURSE 


I    f 


BOOK    I. 
rilE    ALGEBRAIC    LANGUAGE, 


CHAPTER    I. 

OF    ALGEBRAIC    NUMBERS    AND    OPERATIONS. 


General   Definitions. 

1.  Definition.  Mathematics  is  the  science  which 
treats  of  the  relations  of  magnitudes. 

The  magnitudes  of  matliematics  are  time,  space,  force, 
vahie,  or  other  things  which  can  be  thought  of  as  entirely 
made  up  of  parts. 

3.  Bef.  A  Quantity  is  a  definite  portion  of  any 
magnitude. 

Example.  Any  definite  number  of  feet,  miles,  acres, 
bushels,  years,  pounds,  or  dollars,  is  a  quantity. 

3.  Def.  Algebra  treats  of  those  relations  which 
are  true  of  quantities  of  every  kind  of  magnitude. 

4.  The  relations  treated  of  in  Algebra  are  discovered 
by  means  of  numbers. 

To  measure  a  quantity  by  number,  we  take  a  certain  por- 
tion of  the  magnitude  to  be  measured  as  a  unit,  and  express 
how  many  of  the  units  the  quantity  contains. 

Remark.  It  is  obviously  essential  that  the  quantity  and 
its  unit  shall  be  the  same  kind  of  magnitude. 

5.  Def.  A  Concrete  Number  is  one  in  which  the 
kind  of  quantity  which  it  measures  is  expressed  or 
understood  ;  as  7  miles,  3  days,  or  10  pounds. 


THE  ALGEBRAIC   LANGUAGE. 


C>.  Def.  An  Abstract  Number  is  one  in  which  no 
particular  kind  of  unit  is  expressed  ;  as  7,  3,  or  10. 

lluMAiiK.  An  al)stract  number  may  be  considered  as  a 
concrete  one  expressing  a  certain  number  of  u.iits,  ivitliout 
respect  to  the  kind  of  units.     Thus,  7  means  7  units. 

Algebraic  Numbers. 

7.  In  Arithmetic,  the  numbers  begin  at  0,  and  in- 
crease without  limit,  as  0,  1,  2,  3,  4,  etc.  But  the 
quantities  we  usur^ly  measure  by  numbers,  as  time 
and  space,  do  not  reully  begin  at  any  point,  but  extend 
without  end  in  opposite  directions. 

For  example,  time  has  no  beginning  and  no  end.  An 
epoch  of  time  1000  years  from  Christ  may  be  cither  1000  years 
after  Christ,  or  1000  years  before  Christ. 

A  heavy  body  tends  to  fall  to  the  ground.  A  body  which 
did  not  tend  to  move  at  all  when  unsupported  would  have  no 
weight,  or  its  weight  would  be  0.  If  it  t..nded  to  rise  upward, 
like  a  balloon,  it  would  have  the  opposite  of  Aveight. 

If  we  have  to  measure  a  distance  from  any  point  on  a 
straight  line,  \\<i  mp.y  measure  out  in  either  direction  on  the 
line.     If  the  one  direction  is  east,  the  other  will  be  west. 

One  who  measures  his  wealth  is  poorer  by  all  that  he  owes. 
If  he  (  wes  more  than  he  possesses,  he  is  worth  less  than 
nothing,  and  there  is  no  limit  to  the  amount  he  may  owe. 

8.  In  order  to  measure  such  quantities  on  a  uni- 
form system,  the  numbers  of  Algebra  are  considered  as 
increasing  from  0  in  two  opposite  directions.  Those  in 
one  direction  are  called  Positive;  those  in  the  other 

direction  Negative. 

9.  Positive  numbers  are  distinguished  by  the  sign 
-f,  plus  ;  negative  ones  by  the  sign  — ,  minus. 

If  a  positive  number  measures  years  after  Christ,  a  negative 
one  will  mean  years  before  Christ. 

If  a  positive  number  is  used  to  measure  toward  the  right,  a 
negative  one  will  measure  toward  the  left. 


ALOEBRAIC  NUMBElta. 


L  wliicli  no 
or  10. 

iilcrcd  as  a 
its,  ^vithout 
nits. 


0,  and  in- 
But  the 

^s,  as  time 
but  extend 

0  end.     An 
r  1000  years 

body  whicli 
luld  have  no 
rise  upward, 
it. 

point  on  a 
ction  on  the 
e  west, 
that  he  owes, 
bh  less  than 
lay  owe. 

i  on  a  uni- 
msidered  as 

1.  Those  in 
a  the  other 

by  the  sign 
mis. 

ist,  a  negative 

d  the  right,  a 


If  a  positive  numl)cr  measures  weight,  the  negative  ouo 
will  imply  levity,  or  tehdency  to  rise  from  the  earth. 

IE  a  positive  nnniber  measures  property,  or  credit,  the  nega- 
tive one  w  ill  imi)ly  debt. 

10.  The  scries  o^  algebraic  numbers  will  therefore 
be  considered  as  arranged  in  the  following  way,  the 
series  going  o'lt  to  infinity  in  both  directions. 


-S»    NEGATIVE  DIRECTION. 

Before. 
Downward, 
Debt, 
etc. 


POSITIVE   DIRECYION.     f^" 

After. 
Upward. 
Credit, 
etc. 


etc.  -5,  -4,  -3,  -2,  -1,  0,  +1,  +2,  +3,  +4,  +5,  etc. 

Rem.  It  matters  not  which  direction  we  take  as  the 
positive  one,  so  long  as  we  take  the  opposite  one  as 
negative. 

If  we  take  ;me  before  as  positive,  ti!he  after  will  be  nega- 
tive ;  if  we  take  west  as  the  positive  direction,  east  will  be 
negative;  if  we  take  debt  as  positive,  credit  will  be  negative. 

11.  Positive  and  negative  numbers  may  be  conceived 
as  measuring  distances  from  a  fixed  point  on  a  straight 
line,  extending  indefinitely  in  both  directions,  the  dis- 
tances one  way  being  positive,  and  the  other  way 
negative,  as  in  the  following  scheme :  * 

etc.   — 7, —6, —5, -4, -3. -2, -1,     0,  +1. +2, +3,  +4, +5,  +6.  +7,  etc. 

I       i       I       I       I       I       I       I       I       i       I       I       I       I       I 

In  this  scale,  the  distance  between  any  two  consecu- 
tive numbers  is  considered  a  unit  or  unit  step. 

13.  Def.  The  signs  +  and  —  are  called  the  Alge- 
braic Signs,  because  they  mark  the  direction  in  which 
the  numbers  following  them  are  to  be  taken. 

*  The  student  sliould  copy  this  scale  of  numbers,  and  have  it  before 
him  in  studying  the  present  chapter. 


6 


THE  ALGEBRAIC  LANGUAGE. 


The  sign  4-  may  bo  omitted  before  positive  mimbers,  Avhcn 
no  jimbigiii'y  is  thius  produced.  The  mimbcrs  'I,  6,  Vi,  taken 
ulone,  signify  4-3,  +5,  4-1;:^.  But  the  negative  sign  must 
always  be  written  wiien  a  negative  number  is  intended. 

13.  Def.  One  number  is  said  to  be  Algebraically- 
Greater  tlian  another  when  on  tlie  preceding  scale  it 
lies  to  the  positive  (right  hand)  side.    Thus,  \ 

—  2    is  algebraically  greater  than    —  7 ; 
0    "  "  .   "  "        -2; 

6    "  "  '  «  «        -5. 


Algebraic  Addition. 

14.  Def.  In  Algebra,  Addition  means  the  combi- 
nation of  quantities  according  to  their  algebraic  signs, 
tlie  positive  quantities  being  counted  one  way  or  added, 
and  negative  ones  the  opposite  way  or  subtracted. 

15.  Def.  The^lgebraic  Sum  of  several  quantities 
is  the  surplus  of  the  positive  quantities  over  the  nega- 
tive ones,  or  of  the  negative  quantities  over  the  positive 
ones,  according  as  tlie  one  or  the  other  is  the  greater. 

The  sum  has  the  same  algebraic  sign  as  the  prepon- 
derating quantity. 


Example. 

Tlie  sum  of 

+  7    and 

-7    is 

0; 

+  9       " 

_7     " 

+  2; 

+  5       " 

_7    " 

—  2. 

The  sum  of  several  positive  numbers  may  be  represented 
on  the  line  of  numbers,  §  11,  by  the  length  of  the  line  formed 
by  placing  the  lengths  represented  by  the  several  numbers 
end  to  end.  The  total  length  will  be  the  sum  of  the  partial 
lengths. 

If  any  of  the  numbers  are  negative,  the  algebraic  sum  is 
represented  by  laying  their  lengths  off  in  the  opposite  direction. 

Example  1.  The  algebraic  sum  of  the  four  numbers  9, 
—  7,  1,  —  G,  would  be  represented  thus : 


J 


t 


imbcrs,  wJien 
5,  1:;*,  taken 
e  sign  must 
nded. 

ebraically 

Liig  scale  it 

-7; 
-2; 
-5. 


the  combi- 
)raic  signs, 
'  or  added, 
icted. 

quantities 
'  the  nega- 
le  positive 
greater. 
le  prepon- 


•epresented 
ine  formed 
1  numbers 
the  partial 

'aic  sum  is 
)  direction. 

umbers  9, 


I 


4 


ALGEBRAIC  ADDITION. 


+  9 


ET 


Here,  starting  from  0,  we  measure  9  to  tJie  right,  then  7 
to  the  left,  tlien  1  to  the  riglit,  then  G  to  the  left  The  result 
■would  be  3  steps  to  the  left  from  0,  that  is,  —  3,  Thus,  —  3 
is  the  algebraic  sum  of  +9,  —7,  +1,  and  — 6. 

Ex.  2.  If  we  imagine  a  person  to  walk  back  and  forth 
along  the  line  of  num!  jrs,  his  distance  from  the  starting- 
point  will  ahvays  be  the  algebraic  sum  of  the  separate  distances 
he  has  walked. 

Ex.  3.  A  man's  wealth  is  the  algebraic  sum  of  his  posses- 
sions and  credits,  the  debts  which  lie  owes  being  negative 
credits.  If  he  has  in  money  SIOOO,  due  from  A  11200,  due  to 
X  1500,  due  to  Y  $350,  his  possessions  would,  in  the  language 
of  algebra,  be  summed  up  as  follows : 

Cash,  .        .        .        .         +  $1000 

Due  from  A,  .....+    1200 

Due  from  X,  ....        —      500 

Due  from  Y,  ....        —      350 

Sum  total,    .        .        .        .        +  $1350 

[In  the  language  of  Algebra,  the  fact  that  he  owes  X  $500 
may  be  expressed  by  saying  that  X  owes  him  —  $500.] 

16.  Def.  To  distinguish  between  ordinary  and 
algebraic  addition,  the  former  is  called  Numerxced  or 
Arithmetical  addition. 

Hence,  the  numerical  sum  of  several  numbers 
means  their  sum  as  in  arithmetic,  without  regard  to 
their  signs. 

17.  Rem.  In  Algebra,  whenever  the  word  su??i 
is  used  without  an  adjective,  the  algebraic  sum  is 
understood. 


8 


rilE  AfMKBRAIC  LANGUAGE. 


Alfjobrjiic    Subtraction. 

18.  Mnnnraiultim  of  arithmcliral  d'Jlnitions andopcraliuns. 
'I'lio  Subtrahend  is  the  (|iiiuitity  tj  be  subtnicted. 

'I'lie  Minuend  is  tlic  (iiuintitv  ^"^ni  which  the  subtnihcnd 
is  taken. 

The  Remainder  or  Difference  is  whiit  is  left. 

If  we  subtract  4  from  7,  the  remainder  3  is  tlie  number  of 

unit  steps  on  the  scale  of  numbers  (§  11)  from  +4  to  +7. 

Tbirf  is  true  of  any  arithmetical  ditl'erenee  of  numbers.     In 

Algebra,  the  oi)eration  is  generalized  as  fc^llows: 

19,  Dcf.  The  Algebraic  Difference  of  two  num- 
bers is  represented  by  the  distance  from  one  to  the 
other  on  the  scale  of  numbers. 

The  number  from  which  we  measure  is  the  Subtra- 
hend. 

That  to  which  we  m(\isure  is  the  Minuend. 

If  the  minuend  is  algebraically  the  greater  (§  13), 
the  difference  is  positive. 

If  the  minuend  is  less  than  the  subtrahend,  the  dif- 
ference is  negative. 

In  Arithmetic  we  cannot  subtract  a  greater  number  from  a 
less  one.  But  there  is  no  such  restriction  in  Algebra,  because 
algebraic  subtraction  does  not  mean  taking  away,  but  finding 
a  ditference.  However  the  minuend  and  subtrahend  may  bo 
situated  on  the  scale,  a  certain  number  of  spaces  toward  the 
right  or  toward  the  left  will  always  carry  us  from  the  subtra- 
hend to  the  minuend,  and  these  sjjaces  make  \\\}  the  difference 
of  the  two  numbers. 

30.  The  general  rule  for  algebraic  subtraction  may  be 
deduced  as  follows :  It  is  evident  that  if  we  pass  from  the 
subtrahend  to  0  on  the  scale,  and  then  from  0  to  the  minuend, 
the  algebraic  sum  of  these  two  motions  will  be  the  entire  space 
between  the  subtrahend  and  minuend,  and  will  therefore  be 
the  remainder  required.  But  the  first  motion  will  be  equal  to 
the  subtrahend,  but  ]K)sitive  if  that  quantity  is  negative,  and 
vice  versa,  and  the  second  motion  will  be  equal  to  the  minuend. 


4 


ALGEBRA  IC  MUL  TIPLICA  TION. 


9 


Hence  the  rcnuiiiKlcr  will  be  found  by  c'han<(iii,c:  the  al<^('bniio 
sign  of  the  subtruhoud,  and  thou  adding  it  algebraically  to  tho 
mi'uicnd. 

EXAMPLES. 

Subtracting  +5  from  +  8,  the  ditlerence  is      8  —  5  =  3. 


+  8 

+  5, 

«<      5  _  8  =  -  3. 

+  8 

-5, 

<'  —  5  — 8  =  -13. 

-8 

5, 

*'       5  -f  8  =  +  13. 

+  13 

0, 

"                      -  13. 

-13 

0, 

+  13. 

21.  By  comparing  algebraic  addition  and  subtraction,  it 
will  bo  soon  that  to  subtract  a  positive  number  is  the  same 
thing  as  to  add  its  negative,  and  vice  versa.     Thus, 

To  subtract  5  from  8  gives  the  same  result  as  to  add  —  5 
to  8,  namely  3. 

To  subtract  —  5  from  8  gives  8  -f-  5,  namely  13. 

Hence,  algebraic  subtraction  is  equivalent  to  the 
algebraic  addition  of  a  number  with,  the  oi)posite 
algebraic  sign.  Algebraists,  therefore,  do  not  consider 
subtraction  as  an  operation  distinct  from  addition. 

Algebraic   ^lultipHcatioii, 

22.  Memorandum  of  arithmetical  definitions. 

The  Multiplicand  is  the  quantity  to  be  multiplied. 
The  Multiplier  is  the  number  by  which  it  is  multiplied. 
Tlio  result  is  called  the  Product. 

Factors  of  a  number  are  the  multiplicand  and  multiplier 
which  produce  it. 

23.  To  multiply  any  algebraic  quantity  by  a  posi- 
tive whole  number  means,  as  in  Arithmetic,  to  take  it  a 
number  of  times  equal  to  the  multiplier. 

Thus,  4x3=      4  +  4  +  4=4-12; 

—  4   <3  =  -4  —  4  —  4=-12. 

The  product  of  a  negative  multiplicand  by  a  positive 
multiplier  will  therefore  be  negative.     .  . 


10 


THE  ALOKDRAIC   LANGVAUE. 


24.  If  the  multiplier  is  nop^tativo,  the  sign  of  tlio 
product  will  be  the  opposite  of  what  it  would  be  if  tiie 
iimltiplior  were  positive. 

Tims,  +4  X  -3  =  -12; 

'—4  X  -3  =   4-  1^.^ 
The  product  of  two  negative  factors  is  therefore 
positive. 

25.  Tlio  most  simple  way  of  mjistoriii^  tlio  use  of  jil^t'lu-iiio 
si^ns  in  multiplication  is  to  think  of  tho  ni^^ii  —  us  incaniti;^ 
opposite  in  direction.  Thus,  in  §11,  — t  is  opposite  in 
direction  to  +  •!>  the  direction  being  thiit  from  0.  If  we  mul- 
tiidy  this  negative  factor  by  a  negative  multiplier,  the  direction 
will  he  the  npposi/c  of  negative,  that  is,  it  will  he  positice.  A 
third  negative  factor  will  make  tho  product  negative  again,  a 
fourth  one  positive,  and  so  on.     For  example, 

-3  X  -4  =  +13; 
—2  X  —3  X  — 4  =  -3  X  +12  =  —  24; 
— 3  X  —2  X  -3  X  —4  =  -3  X  -24  =  +  72 ; 
etc.  etc. 

Hence, 

26.  TJicorem,  The  continued  product  of  an  even 
number  of  negative  factors  is  jiositive  ;  of  an  odd  nuni- 
"ber,  negative. 

Rem.  Multiplying  a  number  by  —1  simply  changes 
its  sign. 

Thus,  +4  X  — 1  =  —  4; 

-4  X  -1  =  +  4. 

EXERCISES. 

Find  the  algebraic  sums  of  the  following  quantities : 

1.  4  —  G  +  12  —  1  —  18. 

2.  _  G  —  3  —  8. 

3.  _  6  —  10  —  9  +  34. 

4.  Subtract  the  sum  in  Ex.  3  from  the  sum  in  Ex.  2. 

5.  Subtract  the  sum  5  —  6  +  3  —  1  —  IG,  from  the  sum 


III! 


AiahUJiAIO  DivmoN. 


11 


sum 


6.  Siiblnu't  the  sum  5  —  G  +  3  —  1  —  10,  from  the  sum 
7  _  3  -  8  -f-  4. 

7.  Form  the  i)rofluct  —  7  X  8. 

8.  Form  the  product  —8x7. 

9.  Form  the  product  Gx— 5x7x  — 4. 

10.  Form  the  product  — Gx—  llx8x— 2. 

11.  Form  the  product  — Ix— Ix— Ix—  1. 

12.  Suhtnict  the  8um  in  Ex.  1  from  the  .sum  iu  F].\,  3,  ami 
multiply  the  remiiindcr  hy  the  sum  in  FjX.  'i. 

13.  Suhtnict  8  from  —  J),   —  ;}  from  —  1,  —  1  from  8,  iind 
lind  the  sum  of  the  three  remiiindcrs. 

14.  kSuhtnict  7  from  — 9  and  the  reniiiiiulcr  from  *^,  mul 
multiply  the  result  hy  the  i»roduct  in  Ex. 


<• 


Algebniic    Division. 

27.  Memnrnndnm  of  arithmetical  ihlinitions. 

The  Dividend  is  the  (pumtity  to  ho  divided. 
The  Divisor  is  the  numher  hy  which  it  is  divided. 
The  Quotient  is  the  result. 

28.  Rule  of  Signs  in  Division.  The  roquircmcnt 
of  division  in  Algebra  is  the  same  as  iu  Arithmetic  ; 
namely, 

Tlie  product  of  the  quotient  hy  the  divisor  must  he 
cqii(d  to  the  dividend. 

In  Algehra,  two  quantities  are  not  equal  unless  they  have 
tliG  same  algehraic  sign.     Therefore  the  product, 

quotient  x  divisor 

must  have  the  same  algehraic  sign  as  the  dividend.     From 
this  we  can  deduce  the  rule  of  signs  in  division. 

Let  us  divide  G  hy  2,  giving  G  and  3  both  algebraic  signs, 
and  find  the  signs  of  the  quotient  3 : 

+  3x+2=+G;  therefore,  -f  G  divided  by  +2  gives  +3. 

+  3  X  — 2  =  — G;          *'  — G       ''        "  —2     "      +3. 

-3x+2=— G;          "  — G       **        *'   +2     "      -3. 

-3  X  — 2  =  4-G;          "  -f-G      "        "  —2     "     -3. 


12 


i  IE  ALGEBRAIC  LANGUAGE. 


'  ncnce,  tlie  rule  of  signs  is  the  same  in  division  as  in  mul- 
tiplication, namely  : 

Like  si^ns  in  dividend  and  divisor  give  +.     Unlike 


signs  give  — . 


EXERCISES, 


Expcnte  the  following  algebraic  divisions,  expressing  each 
result  as  a  whole  number  or  vulgar  fraction : 

Dividend,  —  T  +  10  —  11  +  25  ;  divisor,  20  —  3. 
Dividend,  12  —  3  +  15  —  10  ;       divisor,  3  —  10. 
Dividend,  25  —  3G  +  6  —  20  ; 
Dividend,  —  7  x  —  8  ; 
Dividend,  50  +  8  x  —  3  ; 
Dividend,  —  24  x  —  1 ; 


I. 

2. 

3- 

4. 

5- 
6. 

7- 


divisor,  —  3  +  8. 

divisor,  —8  +  4. 

divisor,  —  4  —  4. 

divisor,  —  3  x  —  3. 


Dividend,  —13  x  —10  x  -— 8 ;  divisor,  — 4x5x— G. 


8.  Divie:  nd,  —  1  x  —  1 ; 


divisor,  —  3  x  —  3. 


ha 

US( 

syi 
eh 


syi 

in 

nil 


■•♦»♦■■ 


CHAPTER    II. 
ALGEBRAIC     SYM  BOLS. 


Symbols  of  Quantity. 

29.  Alfi:ebraic  quantities  may  be  represented  by 
letters  of  the  alphabet,  or  other  characters. 

The  characters  of  Algebra  are  called  Symbols. 

30.  Def.  The  Value  of  an  algebraic  symbol  is  the 
quantity  which  it  rejiresents  or  to  which  it  is  equal. 

The  value  of  a  symbol  may  be  any  algebraic  quan- 
tity whatever,  positive  or  negative,  which  we  choose  to 
assign  to  the  symbol. 

31.  The  language  of  Algebra  differs  in  one  respect  from 
ordinary  language.     In  the  latter,  each  special  word  or  sign 


I 


SIGJffS   OF  OPERATION. 


n 


has  a  definite  and  invariable  meaning,  which  every  one  who 
uses  the  language  must  learn  once  for  all.  But  in  Algebra  a 
symbol  may  stand  for  any  quantity  which  the  writer  or  speaker 
chooses,  and  his  results  must  be  interpreted  according  to  this 


mean  nig. 


33.  The  same  character  may  be  used  to  represent  several 
quantities  by  applying  accents  or  attaching  numbers  to  it  to 
distinguish  the  different  quantities.  Thus,  the  four  symbols, 
a,  a,  a",  a'",  may  represent  four  different  quantities.  The 
symbols  rt'if  a.,,  a.^,  c(i,  a^,  etc.,  maybe  u.«ed  to  designate  any 
numl)er  of  (piantities  which  are  distinguished  by  the  small 
number  written  after  the  letter  a. 

Signs  of  Operation, 

33.  In  Algebra,  tlie  signs  +,  — ,  and  x  are  used, 
as  in  Aritlimetic,  to  represent  addition,  subtraction,  and 
multipllcaiion,  these  operations  being  algebraic,  not 
numerical. 

34.  Sf'r/ns  of  Addition  and  Subtraction.  The  com- 
bination a-\-h  means  the  algebraic  sum  of  the  quantities 
a  and  h,  and  a  —  h  means  their  algebraic  difference. 


EXAMPLES. 

If  rtj  =  +  4  and  J  =  +  3,  then  a-^h=  +7,  a—h  =4-1. 

If  «  =  +  5  and  b  =  —  7,  then  a-\-h=  —3,  a—h  =  +12. 

If  a  =  —  G  and  Z»  =  +  3,  then  a  +  b=  —3,  a—h  =  —9. 

If  a=  —  0  and  b  r=  —  3,  then  «-f-(5>  =  —9,  a—h  =  —3. 

The  signs  of  addition  and  sul)traction  are  the  same  as  those 
nsed  to  indicate  positive  and  negative  quantities,  but  the  two 
applications  may  be  made  without  confusion,  because  the 
opposite  positive  and  negative  directions  correspond  to  the 
opposite  operations  of  adding  and  subtracting. 

35.  Sign  of  Multiplication.  The  sign  of  multipli- 
cation, X ,  is  generally  omitted  in  Algebra,  and  when 
different  symbols  are  to  be  multiplied,  the  multiplier  is 


14 


TUE  LANGUAGE   OF  ALGEBRA. 


written  before  the  multiplicand  witliout  any  sign  be- 
tween them. 

Thus,  4a    means    a  x  ^. 

ax        "         X  X  a. 
^ahmy        "         y  x  m  x  b  X  a  x  3. 

If  numbGrs  are  used  instead  of  symbols,  some  sign  of  mul- 
ti[)lication  must  be  inserted  between  them  to  avoid  confusion. 
Thus,  34  would  be  confounded  with  the  number  thirtij-fuur. 
A  simple  dot  is  therefore  inserted  instead  of  the  sigu  x . 

Thus,  3-4  =  4x3  =  13. 

3-12.2  =  72. 
1-2-3-4-5  =  120. 
1.2-3. 4. 5-6  =  720. 
The  only  reason  why  the  poiut  is  used  instead  of   x,  is 
that  it  is  more  easily  written  and  takes  up  less  space. 

36.  Division  in  Algebra  is  sometimes  represented 
by  the  symbol  -r-,  the  dividend  being  placed  to  the  left 
and  the  divisor  to  the  right  of  this  symbol. 

Ex.     a  -v-h  means  the  quotient  of  a  di"ided  by  h. 

But  division  is  more  generally  represented  by  writing 
the  dividend  as  the  numerator  and  the  divisor  as  the 
denominator  of  a  fraction. 

Ex.     The  quotient  of  a  divided  by  l  is  written  y- 

It  is  shown  in  Arithmetic  that  a  fraction  is  equal  to  the 
quotient  of  its  numerator  divided  by  its  denominator  ;  hence 
this  expression  for  a  quotient  is  a  vulgar  fraction. 

37.  Powers  and  Exponents.  A  Power  of  a  quan- 
tity is  the  i)roduct  obtained  by  taking  that  quantity  a 
certain  number  of  times  as  a  factor. 

Def.  The  Degree  of  the  power  means  the  number 
of  times  the  quantity  is  taken  as  a  factor. 

If  a  quantity  is  to  be  raised  to  a  power,  the  result 
may,  in  accordance  with  the  rule  for  multiplication,  be 


4 


exy 
of 


an( 


SIGIfS   OF  OPERATION. 


15 


X,   IS 


expressed  by  writing  the  quantity  the  required  number 
of  times. 

Examples.     The  fifth  power  of  a  may  be  written 
nxaxaxaxa    or    aaaaa ; 
and  the  fourth  power  of  7,        7-7-7.7  =  2401. 

To  save  repetition,  the  symbol  of  which  the  power  is 
to  be  expressed  is  written  but  once,  and  tlm  number  of 
times  it  is  taken  as  a  factor  is  written  in  small  figures 
after  and  above  it. 


Thus, 


is  written 


^4  . 


<( 


<l 


x\ 


aaaaa 

7.7.7.7 

XXX 

Def.  A  figure  written  to  indicate  a  power  is  called 
an  Exponent. 

D(f.  The  operation  of  forming  a  x:)ower  is  called 
Involution. 

38.  Boots.  A  Root  is  one  of  the  equal  factors 
into  which  a  number  can  be  divided. 

Def.  Tlie  figure  or  letter  showing  the  number  of 
equal  factors  into  which  a  quantity  is  to  be  divided  is 
called  the  Index  of  the  root. 

The  square  root  of  a  symbol  is  expressed  by  writing 
the  sign  ^/  (called  root)  before  it. 

Ex.  I.     V49  means  the  square  root  of  49,  that  is,  7. 
Ex.  2.     VaJ    means  the  square  root  of  x. 

Any  other  root  than  the  square  is  represented  by 
writing  its  index  before  the  sign  of  the  root. 

Ex.  I.     v^ic  means  the  cube  root  of  a:. 
Ex.  2.     ^/x  means  the  fourth  root  of  a;. 

Def.  The  operation  of  extracting  a  root  is  called 
Evolution. 

39.  The  operations  of  Addition,  Subtraction,  Multi- 
plication, Division,  Involution,  and  Evolution,  are  the 
six  fundamental  operations  of  Algebra. 


16 


TEE  ALGEBRAIC  LANGUAGE. 


40.  Def.  An  Algebraic  Expression  is  any  combi- 
nation of  algebraic  symbols  made  in  accordance  with 
the  foregoing  principles. 


EXERCISES. 

In  the  following  expressions,  suppose 

9,  and  com 


0,  m  =  3,  11=^  4,  2^ 


«  =  -  7,  J  =  -  5, 
pute  their  numerical 


Yahies. 


I.  a  ■\-  h  -\-  m  ■{-  p. 
3.  m  —  n  —  a  —  b. 
5.     3a  —  m  ■\-  h  —  2fi. 


2.     a  +  m  +  n. 

4.     n  +  p  —  m  —  a. 

6.     'Za  —  7p  +  2b  —  m. 


7. 

?>)}  np. 

8, 

mncp. 

9- 

bmn. 

10. 

bnp. 

II. 

ahmp. 

12. 

2'^abnp. 

IS- 

am -\-  1)71. 

14. 

am  —  b)i. 

IS- 

hp  —  ayi. 

16. 

Q>p  +  an. 

17- 

n'^p  +  7)1^1), 

18. 

m^n  —  ap^. 

19. 

a^  +  V^' 

20. 

«3  +  ¥. 

21. 

«3  _  hK 

22. 

a^m  —  b^n. 

23- 

aW  —  m^n\ 

24. 

a%-^  —  Wm\ 

25. 

ab^  +  a^. 

26. 

ab^  —  a%. 

27. 

ab  +  mn 

• 

ab  —  7nn 

28. 

ac  —  bp 
b)i  —  vip 

2mhi^  —  10^3 

ab  —  mp 

29. 

, « 

p  —  bcm 

30. 

m  —  n 

In  the  following  expressions,  suppose  «  =  8,  J  =  —  3,  and 
X  to  have  in  succession  the  fifteen  values  — 7,  — 6,  — 5,  etc, 
to  +7,  and  compute  the  fifteen  corresponding  values  of  each 
expression : 

a  +  bx 


-ft- 


31.    x^  -{-bx  ■{•  a.                  32.    

bx 

Arrange  the 

results  in  a  table,  thus  : 

a;  =  -7; 

Expression  31  =  78  ; 

Exp.  32  = 

X  z=  —Q; 

«            "  =  62  ; 

etc. 

X  =  —5  ; 

"            «  =  48. 

etc. 

etc.               etc. 

4 

i 


the 


T  combi- 
ice  witli 


umerical 


m. 


3,  and 

5,  etc, 
)f  each 


-w 


1 


I 


COMPOUND   EXPREbSIONS. 

CHAPTER     III. 

FORMATION    OF    COMPOUND    EXPRESSIONS. 


17 


Fuiidaiiieiittil   Principles. 

41.  Tlie  following  are  two  fundamental  principles  of 
the  algebraic  language : 

First  Principle.  Every  algebraic  expression,  how- 
ever complex,  represents  a  quantity,  and  may  be 
operated  upon  as  if  it  were  a  single  symbol  of  that 
quantity. 

Second  Principle.  A  single  symbol  may  be  used 
to  represent  any  algebraic  expression  whatever. 

43.  When  an  expression  is  to  be  operated  ux)on  as 
a  single  quantity,  it  is  enclosed  between  pan^nthesc^s, 
but  the  parentheses  may  be  omitted,  when  no  ambiguity 
or  error  will  result  from  the  omission. 

Example.  Let  us  have  to  subtract  h  from  a,  and  mnlti})ly 
the  remainder  by  the  factor  m.  The  remainder  will  be  ex- 
pressed by  ffi  —  h,  and  if  we  w'rite  the  product  of  this  quantity 
by  m,  in  the  way  of  §  35,  the  result   all  be 

ma  —  5. 
But  this  will  mean  h  subtracted  from  mn,  which  is  not  what 
we  want,  because  it  is  not  a,  but  a  —  b  which  is  to  be  muUi- 
plied  by  m.     To  express  the  required  operations,  we  enclose 
a  —  h  in  brackets  or  parentheses,  and  write  m  outside,  tLi;s  : 

m  (a  —  I)). 


NUMERICAL     EXAMPLES. 

7(8-2)  =  7-G  =  42;    but    7-8-2  =  50  —  2 

12(3  +4)  =  12-7  =  84. 

(G  +3)  (2  +  G)  =  9.8  =  72. 

(7 -4)  (1-5)  (2+  7)  =  3  X  -4-9  =  -108. 

2 


=  54. 


18 


TUE  LANGUAOE  OF  ALGEBRA. 


Example  2.  Suppose  lliat  the  expression  a  —  b  -\-  c  is  to 
be  added  to  m,  siibtracled  Irom  m,  iiuiltli)Uod  by  m,  divided 
by  m,  raised  to  the  third  power,  or  have  the  cube  root  extracted. 
The  results  will  be  written: 


Added,  ^o  m, 
Subtracted  from  m, 
Multiplied  by  m, 

Divided  by  w, 

Cubed, 

Cube  root  extracted. 


m  -\-  {a  —  b  -{-  6'). 
m—  {a  —  b  -\-  c). 
7)1  {a  —  b  +  c). 
{a  —  b-\-c) 

{a  —  b  +  cy. 
\^{a  —  b  +  c). 


There  are  two  of  these  six  cases  in  which  the  parentheses 
are  unnecessary,  although  they  do  no  harm,  namely,  addition 
and  division,  because  in  the  case  of  addition, 

m  +  [a  —  b  -\-  c) 
is  the  same  as  m  -\-  a  —  b  -{■  c. 

[For  example,     10  +  (8  -  5  +  4)  =  10  +  7  =  17, 
and  10  +  8  —  5  +  4     =  17  also.] 

Again,  in  the  case  of  the  fraction,  it  will  be  seen  that  it  has 
exactly  the  same  meaning  with  or  without  the  parentheses. 

43.  An  algebraic  expression  having  parentheses  as 
a  part  of  it  may  be  itself  enclosed  in  parentheses  with 
other  expressions,  and  this  may  be  repeated  to  any 
extent.  Each  order  of  parentheses  must  then  be  made 
larger  or  thicker,  or  different  in  shape  to  distinguish  it. 

Examples,  i.  Suppose  that  we  have  to  subtract  a  from 
b,  the  remainder  from  c,  that  remainder  from  d,  and  so  on. 
We  shall  have, 

Eirst  remainder, 

Second, 

Third, 

Fourth, 

Fifth,  /. 


b  —  a. 
c  —  {b  —  a), 
d  —[c  —  {b  —  a)], 
e  —  \d  —  [c  —  {b  —  a)]\. 
b-\d-[c-{b-aml 


qu| 
ret 


W 

wi| 


•v  an 

i  fr( 


to 


DEFINITIONS. 


19 


'^  +  ^  is  to 
>»,  divided 
t  extracted. 


aren  til  OSes 
',  uddition 


lat  it  has 
leses. 

teses  as 
5es  with 
to  any 
e  made 
uish  it. 

a  from 
so  on. 


I 


2.  Suppose  that  we  have  to  muUiply  the  dilTcrence  of  ilie 
quantities  a  and  b  hy  p  and  subtract  the  product  from  m.  The 
result  or  remainder  will  be 

m  —p{a  —  li). 

Suppose  now  that  we  have  to  multiply  this  result  by  p-\-q. 
We  must  enclose  both  factors  in  parentheses,  and  the  result 
will  then  be  written  : 

{p  +  q)  [ni  -p  («  -  5)]. 

EXERCISES. 

In  the  following  expression?,  suppose 
a  =  —  1,     i  =  3,    7n  =.  b.     a;  =  —  3,  —  1,  +  1,  +  3, 
and  calculate  the  four  values  of  each  expression  which  result 
from  giving  x  the  above  four  values  in  succession. 

X  {x  —  a){x  —  2a)  {x  —  3a) 


I. 


2. 


1.2. 3-4 
[a{b-x)-b{a-  x)Y 

■ — ■ — ■ m 

m  {b  —  x)  +  b  {m  —  x) 


3- 
4- 


[ax  -\-  b{x  —  ay  -{•m{x  —  aYY 


X  —  m 


X  +  m 
l^{:mx^  ^-b)  —  ^{mx^  —  Z>)]  ^(w^>  —  a). 

Note.     When  the  square  root  is  not  an  integer,  it  will  be  sufficient 
to  express  it  without  competing  it  in  full. 
Thus,  for  ar  =  —  3,  we  shall  have 

^(wx«  +  &)  -  ^{m^  -V)  =  ^748  -  ^/i2■. 

This  is  a  sufficient  answer  without  extracting  the  roots. 

Definitions. 

44.  Coeffi,c'enf.  Any  number  which  multiplies  a 
quantity  is  called  a  Coefl&cient  of  that  quantity.  A 
coefficient  is  therefore  a  multiplier. 

Example.     In  the  expression  ^abx, 

4  is  the  coefficient  of  abxj 
4rt    "  «  «  bx, 


iab 


u 


(( 


X. 


20 


THE  LANGUAGE  OF^ ALGEBRA. 


I)ef.    A  Numerical  Coefficient  is  a  simi)le  number, 
as  4,  in  the  above  c  xaniple. 

Dcf.    A  Literal  Coefficient  is  one  containing  one 
or  more  letters  used  as  algebraic  symbols. 

Rem.     Any  cxuantity  may  be  considered  as  having 
tlie  coefficient  1.  because  \x  is  the  same  as  x. 

Reciprocal.    The  Reciprocal  of  a  number  is  unity 
divided  by  that  number.     In  the  language  of  Algebra, 

1 


Reciprocal  of  ISf  =z 


N 


Formula.  A  Formula  is  an  expression  used  to 
show  how  a  quantity  is  to  be  expressed  or  calculated. 

Term.  When  an  expression  is  made  up  of  several 
parts  connected  by  the  signs  +  or  — ,  each  of  these 
parts  is  called  a  Term. 

Example. — In  the  expression, 

a  +  hx  ■{-  dmj^, 
there  are  tlu'cc  terms,  a,  bx,  and  'dmxK 

When  several  terms  are  enelosed  ])etwoen  parentheses,  so 
as  to  be  operated  on  as  a  single  symbol,  they  form  a  single 
term. 

Thus,  the  expression 

{a  +  ix  -f  3mx^)  (a  +  b) 

forms  but  a  single  term,  though  both  numerator  and  denom- 
inator are  each  a  product  of  several  terms.  Such  expressions 
may  be  called  compound  terms. 

Aggregate.  A  sum  of  several  terms  enclosed  be- 
tween parentheses  in  order  to  be  operated  upon  as  a 
single  quantity  is  called  an  Aggregate. 

Algebraic  expressions  are  divided  into  monomials 
and  polynomials. 

A  Monomial  consists  of  a  single  term. 


i 


% 


to  sj 
ofsil 


pre^ 
etc. 


DEFINITIONS. 


21 


number, 
^ing  oiiG 
I  liavin 


is  unity 
Igobra, 


ised  to 
lated. 

several 
'  these 


'SOS,  so 
single 


3nom- 
ssions 

[  be- 
as  a 

ials 


A  Polynomial  consists  of  more  than  one  term. 
A  Binomial  is  a  polynomial  of  two  terms. 
A  Trinomial  is  a  'polynomial  of  three  tenns. 

Note.  Tlie  last  three  words  are  commonly  applied  only 
to  sums  of  simple  terms,  formed  of  single  symbols  or  [)roduct8 
of  single  symbols. 

Entire.  An  Entire  Quantity  is  one  which  is  ex- 
l~)ressed  without  any  denominator  or  divisor,  as  2,  3,  4, 
etc.  ;  a,  &,  a?,  etc.  ;  2«6,  2m^,  ah  {x  —  y\  etc. 

A  Theorem  is  the  statement  of  any  general  truth. 

45.  OtJier  Algebraic  Signs.  Besides  the  signs  al- 
ready defined,  others  are  of  occasional  use  in  Algebra. 

> ,  the  Sign  of  Inequality,  shows  when  placed  be- 
tween tw^o  quantities,  that  the  one  at  the  open  end  of 
the  angle  is  the  greater. 

Ex.  I.     «  >  5  means  a  is  greater  than  5. 

Ex.  2.  m  <Cx  <C.n  means  x  is  greater  than  m,  hut  less 
than  n. 

: ,  another  Sign  of  Division,  is  placed  between  two 
quantities  to  express  their  ratio. 

Thus,  a  :  b  means  the  ratio  of  a  to  b,,  or  the  quotient  of  a 
divided  by  ^.  " 

.'.  means  Hence,  or  Consequently ;  as, 

a  4-  2  =  5  ;  .'.    a  =  3. 

OD  means  a  quantity  infinitely  great,  or  Infinity. 

,  the  Vinculum,  is  sometimes  placed  over  an 
aggregate  to  include  it  in  one  mass,  in  lieu  of  paren- 
theses. 


Ex.     a  —  b  c  —  d  is  the  same  as  {a  —  b){c  —  d). 

It  is  mostly  used  with  the  radical  sign.    We  often  write 

Va  +  b  -\-  c    instead  of    a/(«  -\-  b  -^  c). 


23 


THE  LANGUAGE  OF  ALUEBUA. 


CHAPTER     IV. 

CONSTRUCTION    OF    ALGEBRAIC    EXPRESSIONS. 

4G.  All  operations  upon  algebraic  quantities,  however 
coni])lo.\,  consist  in  combinations  of  the  elementally  operations 
already  described.  The  result  of  each  single  operation  will  be 
an  aggregate,  a  product,  a  quotient,  or  a  root,  and  every  such 
result  may,  in  subsequent  operations,  be  operated  upon  as  a 
single  symbol.  There  are  only  three  cases  in  Avliich  an  expres- 
sion needs  any  modilication  in  order  to  be  operated  upon, 
namely : 

Case  I.  An  aggregate  must  be  enclosed  in  parentheses,  if 
any  other  operations  than  addition  or  division  arc  to  be  per- 
formed upon  it.     (§  43.) 

Case  II.  When  a  product  is  to  be  raised  to  a  power,  or  to 
have  a  root  extracted,  it  may  be  enclosed  in  parentheses  in 
order  to  show  that  the  operation  extends  to  all  the  factors. 

If  we  take  the  product  ahc,  and  write  an  exponent,  2  for 
instance,  after  it  thus,  ahc^^  it  Avould  upply  only  to  c,  and 
would  mean  a  x  b  x  cK  So  with  the  radical  sign  ;  ^abc 
might  mean  only  ^a  xbxc.  To  indicate  that  the  power 
or  root  is  that  of  the  product  as  a  whole,  we  may  enclose  it 
in  parentheses,  thus  : 

S(iuare  root  of  abc  =  ^(alc). 
Square  of  abc  =  {abcy. 

But  a  root  sign  is  commonly  made  to  include  the  whole 
product  by  simply  extending  a  vinculum  over  all  the  factors 
of  the  product,  thus :    Square  root  of  abc  =  Vcibc. 

Case  III.  If  negative  quantities  are  to  be  multiplied, 
merely  Avriting  them  after  each  other  would  lead  to  mistakes. 
Thus,  the  product  ax  —b  x  —c,  if  written  Avithout  the  x 
sign,  would  bo  a  —  b  ~  c,  and  would  not  mean  a  product  at 
all.     But,  by  enclosing  —b  and  — c  in  parentheses,  we  have 

a{~b){-r), 
which  would  correctly  express  the  product  required. 


bo  c<u 
Ti 
encL'  1 
p-r 
sul)tri 
X  of 
to  a  a 
dilTen 
tracte 
foriuc 
noniii 
Tl 
M 


CONSTRUCriON  OF  ALUKDIIAIC  EXPRESSIONS.      23 


1 
I 


47.  The  following  example  will  show  how  operations  may 
be  eoni))'ne(l  to  any  extent. 

The  quantity  a  is  to  be  subtracted  from  J,  and  the  differ- 
ence multiplied  by  y,  forming  a  product  P.  The  quotient  of 
p  —  r  divided  by  q  is  to  be  multiplied  by  m,  and  the  i)roduct 
subtracted  from  P.  The  ditference  is  to  form  the  numeratnr 
JV' of  a  fraction.  To  form  the  denominator,  b  is  to  be  added 
to  a  and  subtracted  from  it,  and  the  product  Q  of  the  sum  and 
dllTt'renco  formed.  The  (piantity  q  is  to  be  added  to  and  sub- 
tracted from  ;;,  and  the  product  R  of  the  sum  and  dilTcrence 
formed.  The  quotient  of  Q  divided  by  R  is  to  form  the  de- 
iiomiiuitor  of  the  fraction  of  which  the  numerator  is  P, 

The  quantity  h  subtracted  from  a  leaves   b  —  a. 

Multiplying  it  by  y,  the  product  P  is        y  {b  —  a). 

Quotient  oi  p  —  r  divided  by  q 


p  —  r 


Multiplying  it  ])y  m, 


m 


p  —  r 


[If  instead  of  multiplying  the  fraction  as  a  whole  by  m, 
we   had   multiplied   its  numerator,    we   should    have   had    to 

enclose    the  2^  —  ^    ^^    parentheses,  thus:       -    ■ — —•     But 

when  the  multipliir  is  written  at  the  end  of  the  line,  between 
the  terms  of  the  fraction,  as  above,  it  indicates  that  the  frac- 
tion, as  a  whole,  is  multiplied  by  7/?.] 

Subtracting  the  last  product  from/*,  it  is  y(b—a) — m  ~ • 

Adding  b  to  a,  a  -{-  b. 

Subtracting  b  from  a,  a  —  b. 

The  product  Q  of  the  sum  and  difference,  (a  -\-  b){a  —  b). 
The  product  R  o^  p  +  q  hy  p  —  q,  {p  +  q)  {p  —  q)* 

{a  -f  J)  (rt  —  b) 


The  quotient  of  Q  divided  by  i?, 


{p-\-fl){p  —  Q) 


*  Tn  mathematical  languap^e,  when  a  substantive  is  followed  by  a 
symlx)l  ill  tliis  manner,  the  latter  is  used  as  a  sort  of  proper  name  to 
dcsigniito  the  substantive,  so  that  the  latter  can  be  afterward  referred  to 
by  the  letter  without  ambiguity. 

In  the  present  case,  the  cajjital  letters  arc  used  in  accordance  with 
the  second  general  principle,  g  41. 


24 


Till':  LANOUAOE  OF  ALaEBIlA. 


Tlio  fraction  Imving  iVfor  its  numerator  and  this  ([uoticnt 
fur  its  dcnomiiiutor  is 

ij(b  —  a)  —  tn^-~- 
(g  -H  b)  {a  ^^ 

48.  By  the  second  gcncnil  ijriiiciplc,  §  41,  a  sinjijle  sym- 
l)oI  niiiy  bo  written  in  j)lace  of  any  iil^^cbriiic  expression  wimtevor. 
When  several  symbols  indicating  such  expressions  are  com- 
bined, the  orijxinal  expressions  may  bo  substituted  for  them, 
and  bo  treated  in  accordance  with  tiic  lirst  principle. 


a  —  7)X 


EXAMPLES. 

Suppose       P  =  a  -{-  bx ;  Q 

T  z=  X  —  f/;  V  =  mpq 

It  is  required  to  form  the  expression 

PQ  -^TV 

PT—  QV' 

The  answer  is 

(rt  +  Ix)  —^^ {x-  y)  mpq 


I         IK,  \       a  —  hx 

(rt  +  ox)  (a;  —  y) —^ —  mpq 


m 


EXERCISES 

Form  the  expressions : 
I.     P-T. 

7.     ^/{P-T). 
9.      V\ 

VP-  OT 


II. 


13- 


IS. 


Qi  _  7^2 

(P+  T){P-T) 

\Q'ry){Q-V)' 

P^-  T 

^{P  -  Tf 


2.  T-P. 

4.  Q-V. 

6.  ViP-i-T). 

8.  P^T^. 

PT^ 

qv 

{3P-2T) 


10. 


12.        -;;, 


14. 
16. 


2 

—  • 


HP±J'y^ 


1 


i 


n- 


23 


EXERCISK8. 


i^-(<2+r)(c>-r) 


32. 


94 


25 


,8     _^^_-  ^ 


30.       T 


/"+<?» 


(  F  -  y'j  (  K  +  7') 


EXERCISES    IN    ALGEBRAIC    LANGUAGE. 

Tlin  foUowliifj  (lUOHtions  nre  proixwod  to  practice  the  student  in  ox- 
proHsiii^  tii«!  relations  of  ((iiantiticH  in  al'^»'braic  laii^iin^e.  Should  any 
u(  thcTu  otter  diHicultieH,  he  ih  reconunendod  to  HubHtitute  numbers  fur 
the  al^'ebraic  letters,  examine  the  process  by  which  he  |)r()ceedH,  and  then 
ai»iily  the  same  process  to  the  lettJTS  that  ho  applied  to  the  numbers.  No 
Bolutiuns  of  equations  are  ru(juired. 

1.  IIow  many  cents  arc  tlicrc  in  m  dollars  ? 

2.  IIow  numy  dollars  in  tn  cents? 

3.  A  man  had  a  dollars  in  one  ])ockot,  and  b  cents  in  the 
other  ;  how  many  cents  had  he  in  all  ?     IIow  many  dollars  ? 

4.  The  sum  of  the  quantities  a  and  b  is  to  be  multiplied 
by  m.     Exi)ress  the  })roduct,  and  its  scjuarc. 

5.  A  man  having  b  dollars  paid  out  in  dollars  to  one  per- 
son and  n  dollars  to  another.  Exi)ress  what  he  had  left  in 
two  ways  ? 

6.  IIow  many  chickens  at  k  cents  a  piece  can  be  purchased 
for  m  dollars  ? 

7.  A  man  Avalked  from  home  a  distance  of  in  miles  at  4 
miles  an  hour,  and  returned  at  the  rate  of  3  miles  an  hour. 
IIow  long  did  it  take  him  to  go  and  come  ? 

8.  A  man  going  to  market  bought  tomatoes  at  h  cents  per 
pock  and  })otatoes  at  k  cents  a  peck,  of  each  an  equal  number. 
They  cost  him  m  cents.     IIow  many  pecks  of  each  did  he  buy  ? 

9.  How  many  minutes  will  it  require  to  go  a  miles,  at  the 
rate  of  b  miles  an  hour  ? 

10.  A  man  bought  from  his  grocer  a  pounds  of  tea  at  x 
cents  a  pound,  b  pounds  of  sugar  at  y  cents  a  pound,  and  e 
]K)unds  of  coffee  at  z  cents  a  pound.  How  many  cents  will 
the  whole  amount  to  ?    IIow  many  dollars  ?    IIow  many  mills  ? 

11.  A  man  bought  /  pounds  of  flour  at  m  cents  a  pound, 


THE  ALGEBRAIC   LANGUAGE. 


and  liandefl  the  grocer  an  a:-dollar  bill  to  bo  changed  ?     How 
many  cents  ought  he  to  receive  in  change? 

12.  From  two  cities  a  miles  apart  two  men  started  out  at 
the  same  time  to  meet  each  other,  one  going  m  miles  an  hcur 
and  tlie  other  ii  miles  an  hour.  How  long  before  they  will 
meet?  How  far  will  the  lirst  one  have  gone  ?  J  low  far  will 
the  second  one  have  gone  ? 

13.  A  man  left  his  n  children  a  bonds  worth  x  dollars 
each,  and  b  acres  of  land  wortii  y  dollars  an  acre ;  but  he 
owed  m  dollars  to  each  of  q  creditors.  What  was  each  child's 
share  of  the  estate  ? 

14.  Two  numbers,  x  and  y,  are  to  be  added  together,  their 
sum  multiplied  by  s,  that  product  divided  by  «  +  ^*,  and  the 
quotient  subtracted  from  h.    Express  the  result. 

15.  The  sum  of  the  numbers  p  and  q  is  to  be  divided  by 
the  sum  of  the  numbers  a  and  b,  forming  one  ((uotient.  The 
ditfcrence  of  the  numbers  p  and  q  is  to  be  divided  by  the  dif- 
ference of  the  numbers  a  and  b,  forming  another  quotient. 
The  sum  of  the  two  quotients  is  to  be  multii^lied  by  ?'-f  s. 
Express  the  product. 

16.  The  quotient  of  x  divided  by  a  is  to  be  subtracted 
from  the  quotient  of  y  divided  by  b,  and  the  remainder  multi- 
plied by  the  sura  of  x  and  y  divided  by  the  difference  between 
X  and  y.     Express  the  result. 

17.  The  number  x  is  to  be  increased  by  G,  the  sum  is  to  be 
multiplieil  by  a  +  b,  q  is  to  be  added  to  the  product,  and  the 
sum  is  to  be  divided  by  r  —  s.    Express  the  result. 

18.  A  family  of  brothers  a  in  number  each  had  a  house 
worth  a  thousand  dollars  each.  What  Avas  the  total  value  of 
all  the  houses  in  dollars  ?     What  was  it  in  cents  ? 

19.  A  grocer  mixed  a  pounds  of  tea  wortli  x  cents  a  pound, 
and  b  pounds  worth  y  cents  a  pound.  How  much  a  pound 
was  the  mixture  worth  ? 

20.  x-{-y  houses  each  had  a-\-b  rooms,  and  each  room 
m  +  n  pieces  of  furniture.  How  many  pieces  of  furniture  were 
there  in  all  ? 

21.  In  a  library  wTre  p-\-q  volumes,  each  volume  had  p  +  q 
pages,  each  page  p  +  q  Avords,  and  each  word  on  the  average 
8  letters.  How  many  letters  Avere  there  in  all  the  books  of  the 
library  ? 

22.  A  post-bov  started  out  from  a  station,  travelling  h 
miles  an  hour.  Tliree  hours  afterw\'ird,  nnother  one  started 
after  him,  riding  ui  miles  an  hour.     How  far  Avas  the  first  one 


al 

St; 


A 


EXERCISES. 


27 


ngcd  ? 


IIgw 


arted  out  at 
iiik's  an  hciir 
ore  they  will 
J  low  far  will 

rth  X  dollars 
acre  ;  but  he 
s  each  child's 

jgether,  their 
/  +  b,  and  the 

be  divided  hy 
Liotient.  The 
id  by  the  dif- 
ther  quotient, 
plied  by  r  +  s. 

be  subtracted 
laindcr  multi- 
•ence  between 

e  sum  is  to  be 
duct,  and  the 
lilt. 

had  a  house 
total  value  of 

V 

cents  a  pound, 
nuch  a  pound 

d  each  room 
furniture  were 

ume  had  p-^fj 
n  the  average 
ic  books  of  the 

.,  travelling  h 
er  one  started 
ls  the  first  one 


■'4 


•^ 


ahead  of  the  second  at  the  end  of  x  hofirs  after  the  second 
started  ? 

23.  Two  men  started  to  make  the  same  journey  of  m  miles, 
one  going  r  miles  an  hour,  aiul  the  other  s  miles  an  hour. 
][o\v  much  sooner  will  the  man  going  /•  miles  an  hour  make 
iiis  journey  than  the  one  going  s  miles  an  hour?  How  mueh 
sooner  will  the  one  going  s  miles  an  hour  make  his  journey 
than  the  one  going  ;■  miles  an  hour  ? 

24.  One  train  runs  from  Boston  to  New  York  in  h  hours, 
at  the  rate  of  n  miles  ar.  hour.  IIow  long  will  it  take  another 
train  running  5  miles  an  hour  faster  to  perform  the  journey  'i 

25.  If  a  man  bought  h  horses  for  t  dollars,  and  n  yoke  of 
oxen  for  m  dollars,  how  much  more  did  one  horse  cost  than  one 
yoke  of  oxen  ?  lIow  much  more  did  one  yoke  of  oxen  cost 
than  one  horse  ? 

26.  A  train  making  a  journey  of  ^m  miles  goes  the  first 
half  of  the  way  at  the  rate  of  /•  miles  an  hour,  and  the  second 
hiilf  at  the  rate  of  s  miles  an  hour.  IIow  long  did  it  take  it  to 
go  ?    What  was  the  average  speed  for  the  journey':' 

27.  Two  men,  A  and  li,  started  to  walk  from  Hartford  to 
New  Haven  and  back,  the  distance  between  the  two  fitios 
being  a  miles.  A  goes  p  miles  an  hour  and  B  q  miles  t  n  hour. 
IIow  far  Avill  A  have  got  on  his  return  journey  when  B  reaches 
Hartford? 

28.  A  man  having  Ic  dollars  bought  h  books  at  8G  each. 
IIow  many  books  at  $1  each  can  lie  buy  with  the  balance  of 
his  money? 

29.  A  man  going  to  his  grocer  with  m  dollars,  bought  s 
pounds  of  sugar  at  a  cents  a  pound,  and  /•  pouuds  of  colt'ee  at 
b  cents  a  pound.  How  numy  barrels  of  ilour  at  q  dollars  a 
barrel  can  he  buy  with  the  balance  of  his  money  ? 

30.  A  man  divided  m  dollars  equally  among  a  poor  Chinese 
and  n  dollars  equally  among*  h  orplians.  Two  of  the  Chinese 
and  three  of  the  orplians  put  their  shares  together  and  bought 
X  Bibles  for  the  heathen.     How  much  did  each  Bible  cost  ? 

31.  A  pedestrian  having  agreed  to  wtilk  the  n  miles  from 
Boston  to  Natick  in  h  hours,  travels  the  first  k  hours  at  the 
rate  of  in  miles  an  hour.  At  what  rate  must  he  travel  the 
remainder  of  the  time? 

32.  A  train  having  to  make  a  journey  of  x  miles  in  li  hours, 
ran  for  k  hours  at  the  rate  of  r  miles  an  hour,  and  then  made 
a  stoj)  of  m  minutes.  How  fast  must  it  go  during  the  remain- 
der of  its  journey  to  arrive  on  time  ? 


\<\ 


BOOK    II. 
ALGEBRAIC     OPERA  TIONS 


General   Remarks. 

The  algebraic  expressions  formed  in  accordiince  with  the 
rules  of  the  preceding  book  admit  of  being  transformed  and 
simplitied  m  a  variety  of  ways.  This  transformation  is  eli'ected 
by  operations  which  have  some  resemblance  to  the  arithmetical 
operations  of  addition,  subtraction,  multiplication,  and  division, 
and  which  are  therefore  called  by  the  same  names. 

In  performing  thesf  algebraic  operations,  the  student  is  not, 
as  in  Arithmetic,  seeking  for  a  result  which  can  be  written  in 
only  one  way,  but  is  selecting  out  of  a  great  variety  of  forms  of 
expression  some  one  form  which  is  the  simplest  or  the  best  for 
certain  purposes.  Sometimes  one  form  and  sometimes  another 
is  the  best  for  a  particular  problem.  Hence,  it  is  essential 
that  the  algebraist,  in  studying  an  expression,  should  be  able  to 
see  the  different  ways  in  which  it  may  be  written. 


Definitions. 

49.  Function,  An  algebraic  expression  containing 
any  symbol  is  called  a  Function  of  the  quantity  repre- 
sented by  that  symbol. 

Ex.  I.  The  expression  Zx^  is  a  function  of  x, 

a  function  of  x  and  also  a 


mi                  .       a  -\-  X  . 
2.  The  expression is 

function  of  a. 


When  an  expression  contains  several  symbols,  we  may 
select  one  of  them  for  sptcial  consideration,  and  call  the  ex- 
pression a  function  of  that  particular  one.  Fur  instance, 
although  the  expressions, 


DEFINITIONS. 


29 


ONS. 


ice  with  the 
sfonnc'd  jmd 
on  is  eft'ected 
arithmetical 
and  division, 

udent  is  not, 
be  written  in 
y  of  forms  of 
the  hest  for 
imes  another 
is  essential 
lid  he  uhle  to 


containing 
iitity  repre- 


and  also  a 

ols,  we  may 

call  the  ex- 

'or  instance. 


m  +  iWx, 
contain  other  symbols  besides  oc,  they  are  both  functions 
of  X. 

50.  An  Entire  Function  is  one  in  wliicli  the  quan- 
tity is  used  only  in  the  operations  of  addition,  subtrac- 
tion and  multiplication. 

Example.     The  expressions 

ax  +  y, 
(«2  _  2/2)  ^  —  {I)i  -^.y)x^  —  x  +  d, 

arc  entire  functions  of  x.    But  the  expressions 


ax 


y 


and    3\/^ 


ax  —  y 

are  not  entire  functions  of  .r,  because  in  the  one  x  appears  as 
part  of  a  divisor,  and  in  the  other  its  square  root  is  extracted. 
>  n  entire  function  of  x  can  always  be  expressed  as  a  sum 
CI  terms,  arranged  according  to  the  powers  of  o:  which  they 
contain  as  factors.    The  form  of  the  expression  will  then  be 

A-^  Bx+  Ct?  ^  D:^  ^  Ex^^  etc., 
where  A,  B,  (7,  etc.,  may  represent  any  algebraic  expressions 
wliich  do  not  contain  x. 

51.  Like  Terms  are  those  which  are  formed  of  the 
same  algebraic  symbols,  comliined  in  the  same  way, 
and  differ  only  in  their  numerical  coefficients. 

Ex.    The  terms  ax,  3«.r,  —hax  are  like  terms. 

53.  Tlie  Degree  of  any  term  is  the  number  of  its 
literal  factors. 

Examples.  The  expression  alxy  is  of  the  fourth  degree, 
because  it  contains  four  literal  factors. 

Tlie  expression  :i^  is  of  the  tliird  degree,  because  the  letter 
X  is  taken  three  times  as  a  factor. 

The  expression  ab'^x?'  is  of  the  sixth  degree,  because  it  con- 
tains a  once,  5  twice,  and  x  tiiree  times  as  a  factor. 

When  an  exi)ression  consists  of  several  terms,  its 
degree  is  that  of  its  highest  term. 


hi 


30 


ALGEBRAIC  OrERATIONS. 


1 


CHAPTER     I. 

ALGEBRAIC    ADDITION    AND    SUBTRACTION. 


Algebraic  Atlditioii. 

53.  By  the  language  of  Algebra,  the  sum  of  any  number 
of  quantities,  positive  or  negative,  may  be  expressed  by  Avriting 
them  in  a  row,  Avith  the  sign  +  before  all  the  positive  quan- 
tities, and  the  sign  —  before  the  negative  ones. 

Ex.  A-{-B—D—X-\-  Yy  etc.,  is  the  algebraic  sum  of  the 
several  quantities  A,  B,  —D,  —X,  Y,  etc. 

54.  To  simplify  an  expression  of  tie  sum  of  several 
quantities. 

1.  When  dissimilar  tenns  are  to  be  added,  no  sim- 
plification can  be  effected. 

Ex.  If  we  require  the  sum  of  the  five  expressions,  a,  —xij^ 
mj),  7iq,  and  —hhs,  we  can  only  write, 

a  —  xy  -{■  mp  +  nq  —  hhs, 

according  to  the  language  ot  Algebra,  and  cannot  reduce  the 
expression  to  a  shnpler  form. 

2.  If  mere  numbers  are  among  the  quantities  to  be 
added,  tlieii  algebraic  sum  may  be  formed. 

Ex.  The  sum  of  the  five  quantities  —8,  ab,  5,  mjij),  —15, 
is  found  to  be  —IS  -\-  ab  -\-  mil}}. 

3.  When  several  terms  are  similar,  add  the  coeffi- 
cients and  affix  the  common  symbol  to  the  sum. 

When  no  numerical  coefficient  is  written,  the  coefficient 
-fl  or  —1  is  understood.     (§  44.) 

EXAMPLES. 

a  -\-  a  =.  2a  [because  1  +  1  =  2]. 
2r«  —  a  =  a  [because  3  —  1  =  1]. 


ALGEBRAIC   ADDITION. 


81 


ION. 


ly  numhcr 
by  writing 
tive  qiiaii- 

um  of  the 

)/  several 

[,  no  sim- 

s,  a,  —xij, 

educe  the 
ies  to  be 
np,  —15, 
le  coefR- 
3oefficicnt 


3rt  +  4rt  —  7a  =  0  [because  3  +  4  —  7  =  0]. 

^  _l_  o  t-  —  3^^  —  hx=.  — "Za — 3.T  [adding  the  r^'s  and  the  z's]. 

—  'daxy  -\-  Abm  —  2axi/  +  bm  =  —  5axi/  +  bbm. 

Add  the  expressions, 

1.  Ix  +  5Z»y^  2.<;  —  3b f,  —  4.x  -  5%«,  6x  —  i/,  a-  —  bf. 

WORK. 

7a;  +  5/y?/' 

2x  —  3/// 

—  4  A'  —  5/>//- 

x-  by 

Sinn,  11a:  —  5/^^^ 

2.  8aa:  —  y  —  2?/  +  6,  7ax—y—d  +  am,  2ax—y—'d-\-b2h 
Here  2.r,  «/?«,  and  p,  work. 

8aa:2  —    y  —  2a:  +  5 

—  7r/a^  —   y  —  9  +  «?» 

—  «7,r2  —    y  —  3 +  5;> 


For  convenience,  tlio  several  terms  may  bo 
written  under  each  other,  as  in  the  margin.  The 
coi'llicients  of  X  are  7,  3,  —4,  5,  and  1,  of  which 
the  algebraic  sum  is  11.  Tlie  coefficients  of  .y* 
are  5,  —3,  —5,  —1,  —  1 ;  the  sum  is  —5.  Hence 
the  result. 


all  being  different  sym- 
bols, the  terms  contain 
ing  them  do  noi,  adn  *■■ 
of  siini)lification  (§  54, 
1),  'i'he  numbers  5, 
—9.  —3,  are  added  by 


Sum,    —'Mj  —  2x  —  7  +  am  +  hp 
the  rule  (§  54, 3).     The  coefficients  of  ax^  cancel  each  other  (8—7—1  =  0) 
3.  Add  (S{x-\-  y),  5  {x  +  y)  +  a,  2  {x  +  y)  -  3a. 


Here  the  aggregate,  x  +  y,  enclosed  ia 
parentheses,  is  treated  as  a  simple  symbol. 

Note.  When  the  student  can  add 
the  coeihcients  mentally,  it  is  not  neces- 
sary to  write  the  expressions  under  each 
other.  Nor  is  it  necessary  to  repeat  the 
symbol  after  each  coefficient. 


WORK. 

6  [x  4-  y) 
5  +    a 

_2 -3 

Sum,  13  {x-\-y)—  2a 


EXERCISES. 

1.  3a-\-n  —  ^c^d,3a  —  2b-\-c  —  e,—a  —  b  —  c—d. 

2.  7a  —  {x  +  y),  8a  —  {x  +  y),  3  {x  +  y)  —  IGa. 

3.  7a-2  —  2x  —  5,  2x'^  —  3a:  +'  8,   —  9:^2  +  5a:  +  3. 

4.  x^  ■\-2x  —  y,  4.1-2  _|.  7.C  _  2^,  _  2x^'  -\-x  —  ^y,    —  3a:2 

5.  9  (a  +  bf,  10  (a  +  bf,  {a  +  bf,  2  {a-\-bf,  ^x-y-z. 

6.  2  {m  -f  7z)  +  3  (a  +  b),     {a  +  b)  —  {m  +  n),     (a  +  b) 
—  {m  4-  n). 

•j.     7a3  —  2a2  -f  3aa',  —  a?  —  a^  —  ax,  —  Ga^  +  3a2  —  2ax. 


I 
ii 


32 


ALGEBRAIC   OPERATIONS. 


12. 


13.     - 

14. 
15- 


m       X         m 

6  — ,  4 4  — < 

11        y         n 


8.  (m  +  nf  ■\-x,  2  {m  +  w)^  —  y,  3  (m  +  w)'  —  5ia;, 
(m  4-  nf  —  y. 

9.  (P  +  (if  -  «»  (i»  +  (?)'  +  ^'^  (i'^+'Z)'  +  ^  (;>+^7)Hc. 

10.  Ga  {x  —  y)i  5a  {x  —  y),  2a  {x  —  y),  a{x  —  y). 

11.  2  {m  —  ?0  a;  4  2,  3  (/«  +  ?i)  :c  —  5,  5  {m  +  ?i)  x  -—  G, 
7(»i  -\-  n)x  —  b. 

a'  ''«■*"     6'  a      6'  /^      7'  «~7' 

?_-    2-  —  3—     3- 
y      n'     y        n'     y 

Of  two  fanners,  the  first  had  2a;  —  3y  acres,  and  the 
Bccond  liad  x  —  y  acres  more  than  the  first.  How  many  acres 
had  they  botli  ? 

16.  A  had  %x  dollars,  B  had  y  dollars  less  than  A,  and  C 
had  'iy  dollars  more  than  A  and  B  together.  How  many  had 
they  all  ? 

17.  A  father  gave  his  eldest  son  x  dollars,  his  second  5  dol- 
lars less  than  the  first,  his  third  5  dollars  less  than  his  second, 
and  his  fourth  5  dollars  less  than  his  third.  How  much  did 
he  give  them  all  ? 

55.  Addition  with  Literal  Coefficients.  When  dif- 
ferent terms  contain  the  same  symbol,  multiplied  by 
different  literal  coefficients,  these  coeflScients  may  be 
added  and  the  common  symbol  be  affixed  to  their 
aggregate. 

EXAM  PLES. 

1.  As  we  reduce  the  polynomial 

6a;  +  5a;  —  2a; 
to  the  single  term      (6  +  5  —  2)  a;  =  3a;, 
so  we  may  reduce  the  polynomial 

ax  -{■  bx  —  ex 
to  the  single  term,  {a  ■}-  b  —  c)x, 

2.  The  expression 

mx  -\-  ny  —  bx  -\-  dy  -i-  a  -{■  b 
may  be  expressed  in  the  form 

(/»  —  b)x-\-  {71  -h  d)y  -i-  a  -\-  b. 


SUBTRACTION. 


33 


n)  x  —  Gf 


!,  and  the 
laiiy  acres 

A,  and  0 
many  had 

nd  5  dol- 
is  second, 
much  did 

Tien  dif- 
plied  by 
may  be 
to  their 


? 


EXERCISES. 

Collect  the  coefricicnts  of  x  and  y  in  the  following  ex- 
pressions: 

I.     ax  +  by -\- mx -\- ny. 
mnx  4-  %by  +  p(l^  —  45^. 
Zx  —  2y  4-  Gbx  —  4y  +  7rta:  +  wz  +  ^. 
Sax  +  "sio;  +  iy  +  7^  —  5y  +  a;  —  oy. 
ax  -\-  by  -{-  cz  ■—  rnx  —  ny  —  pz. 
2dx  +  3ey  +  4fz  —  2/>;  —  ^dy  +  ^ez. 
2  3 

2«a;  —  by  —  3bx  —  ^ay. 

1  2,         1         ,   3 

-^ax  +  ^by  - -mx  ^  ^^7iy. 

2  1     ; 

4wa:  4-  2?/  —  3rta;  —  Gere  +  ay  —  -mx  +  ^</.r. 

5f/ia:  —  dm7iy  —  «Ja;  +  4(?<^/?/  —  dx. 
1 


^i 


2. 

3- 
4- 

5- 
6. 

7- 
8. 


lO. 

II. 


12. 

13- 

14. 

15- 
16. 

17. 


3«y  4-  2bx  —  -^dx  4-  2ay  —  3bx. 
\ay-^x  +  2y--ay-hx-\-y. 


3mx  —  ax 


^^ay  +  X  -{■  dx  —  y. 


Zabx  —  my  +  2cV^^—  dy  +  Vx. 
bmV'y  —  Gx  +  Wy  —  SVx  —  y-\-\^f. 
4^^x  —  (jy  +  aVy  +  cx  —  Vy  —  '^(iVy  +  Vx. 

Algebraic  Subtraction. 

50.  Def.  Algebraic  Subtraction  consists  in  ex- 
pressing the  difference  of  two  algebraic  quantities. 

Jiide  of  Subtraction.  It  has  been  shown  (§  21)  that 
to  subtract  a  positive  quantity,  b,  is  the  same  as  to 
adcl,  algebraically,  the  negative  quantity,  -b.  Also, 
that  to  subtract  -b  is  equivalent  to  adding  +b.    Hence 

the  rule : 

Change  the  algebraic  sign  of  all  the  teim-s  of  the 
suUraheud,  or  conceive  them  to  he  changed,  and  then 
pj'oeced  as  in  addition. 


1. 1- 


84 


ALaEUUAW   0PEUAT10N8. 


NUMERICAL      EXAMPLES. 

Min.,    10  +  0  =  10     10+   0  =  10     10-i-   0  =  10     10+   0  = 
Subt,     9       =9       9—  4=  5       9—  8=   1       9  —  12  = 

Kcm.,     1  +  0=  7      1  +  10=11      1  +  U=15      1  +  18^ 


10 
—3 

10 


ALGEBRAIC      EXERCISES. 


I.  From 
Subtract 


ox  —  4ay  +  5/5>  +  c, 
X  —  7aj/  —  8b  i-  d. 

WOBK. 

]\Iiniicncl,  3x  —  Aay  +    5^  +  c 

(Subtrtihcud  with  signs  changed,     —  x  +  7ai/  +    ?b  —  d 

Differenco,  2x  +  '3ai/  +  I'Sb  -\-  c  —  d 

Next  we  may  simply  imagine  the  signs  changed. 

2.  From        7x  —    4bxi/  —  12cj/  +    8J  +  Sac 
Take         2x  +    Uxy  +    Scj/ —    5b  —  2d 

Diff.,   5x  —  llbxy  —   4c//  +  VSb  +  Sac  +  2^/ 

3.  From        Sa  -^  Ob  — 12c  —  ISd  —  4a;  +  dcy 
Take       19(?  —  7^*  —    8c  —  25r/  +  3a;  —  4;/ 

4.  From        257;?  +  201^2  +  92y  +  35«a:  —    0 
Take         140g  —    82z'^  +  20//  +  d2((X  +  14 

5.  From  Sa  +  lib  subtract  Ga  +  205. 

6.  From  «  —  &  +  c  —  d  take  — «  +  Z»  —  c  +  rZ. 

7.  From  Sa  —  2b  +  3c  subtract  4:a  —  Gb  —  c  —  2d. 

8.  From  2x'^  —  Sx  —  1  suljtract  bx'^  —  Gx  +  3. 

9.  From  ix*  —  dx^  —  2x^  —  7x  +  9  subtract  q:^—2x^—2x'^ 
+  7a;  —  9. 

10.  From  2x?  —  2ax  +  Sa^  subtract  x^  —  ax  +  a\ 

11.  From  a^  —  Sa^b  +  3«i2  —  b"^  subtract  —  a^  ^  Sa^b. 

12.  From  7;?:3  —  2.?;3  +  2a;  +  2  subtract  4a;3--2a'2— 2a;— 14. 

13.  From  5  (a;  —  ?/)  +  7  (a;  —  2;)  +  9  {z—x)  take   9  (a;  —  ^) 

+  7  (a;  —  ;?)  +  5  (^  -  a-). 

14.  From  12  {a  —  b)  —  S  {a  +  b)  -{-  7a—  2b  take  7  (a— ^') 
—  5  (a  +  *). 

.X       _  ?/       _  2 


15. 


take  -5^  +  0^ 
y         z         X  y        z 


From  7-  — 11"'  — Id 


7^  +  8^. 
a;         6 


SUBTRACTION. 


35 


Clearing  of  Piirentlicses. 

57.  In  §  42,  2,  it  was  shown  that  an  aggregate  of  terms  in- 
cliulod  between  parentheses  might  be  added  or  subtracted  by 
simi)ly  writing  -f  or  —  before  the  i)aren theses. 

When  an  aggregate  not  multiplied  by  a  factor  is  to  be  added 
or  subtracted,  the  parentheses  may  be  removed  by  tbc  rules 
for  addition  and  subtraction;  as  follows: 

r»8.  Plus  S/f//i  hefore  ParentJieses.  If  the  paren- 
theses are  preceded  hy  the  sign  +,  they  may  be 
removed,  and  all  the  terms  added  without  change. 

Example:  I.    27  +  (8— 5-4  +  7)  =  27  +  8— 5—4  +  7  =  33. 

2.  m  +  {a  —  X  —  y  -\-  z)  z=  m  -{-  a  —  x  —  i/  +  z. 

3.  2x  +  (-  3.*;  -  by)  +  (3?/-4r/)  +  (2y-2«) 

—  2x  —  3x  —  by  -1-  dy  —  4a  +  2y  —  2a 
=  —  X  —  Ga. 

The  sign  +  which  precedes  the  parentheses  should  also  be 
considered  as  removed,  but  if  the  first  term  within  the  paren- 
thesis has  no  sign,  the  sign  +  is  understood,  and  must  be 
written  after  removing  the  parentheses. 

EXERCISES. 

Clear  of  parentheses  and  simplify 
^  —  y  +  i-^'  +  y)' 

^  +  y  ■\-  {y  —  ^)- 

dab  —  2in]j  +  {ab  —  3x  —  2mp). 

2ax  —  3by  +  {vix  —  2ax  — 2^^  +  3i^). 


I. 

2. 

3- 

4- 


«•  3f+(l-«"7)  +  G+^^)- 


59.  Minus  Sign  hefore  Parentheses.  If  the  paren- 
theses are  preceded  by  the  sign  — ,  they  may  be 
removed  and  the  algebraic  sign  of  each  of  the  included 
terms  changed,  according  to  the  rule  for  subtraction  in 
§56. 


5 


I.     27  _  (8  _ 
that  is,  27  —  0  =  21. 


EXAMPLES. 

-4  + 7)  =  27 -8  +  5  +  4-7  =  21; 


ALOEBRAIG  OPERATIONS. 


2.  m  —  {—  a— p  +  y  +  x)  =  m-{-a  +  p  —  y  —  X. 

3.  3rt  +  X  —  (2a  —  bx)  —  (9a;  —  a)  =  3a  4-  a;  —  2a  +  5a; 
—  \)x  +  a. 

Sim  3lifying  as  in  §  54,  this  reduces  to  2a  —  3a;. 

EXERCISES. 

Clear  the  following  expressions  of  parentheses  and  reduce 
the  results  to  the  simplest  form  by  the  method  of  §  54. 

1.  ah  —  {m  —  3ab  +  2aa;)  —  7ab. 

2.  X  —  {a  —  x)  +  (a;  —  a). 

3.  2b  -f  {b  —  2c)  —  {b+  2c). 

4.  ix  —  3y -I- 2z  — {—7x  +  5y —  3z)  —  (x  —  y). 

5.  7aa;  —  2by  —  {Sax  +  3%)  —  (8aa;  —  3^*^). 

6.  (a  —  a;)  —  (a  -I-  a;)  +  2:c. 

7.  —  (a  —  ^i)  —  (J  —  c)  —  (c  —  a). 

8.  —  {3m  -]•  2?i)  —  (3w  —  2n)  +  9m. 

60.  We  may  reverse  the  process  of  clearing  of  parentheses 
by  collecting  several  terms  into  a  single  aggregate,  and  chang- 
ing their  signs  when  we  wish  the  parentheses  to  be  preceded 
by  the  minus  sign.  The  proof  of  the  operation  is  to  clear  the 
parentheses  introduced,  and  thus  obtain  the  original  expression. 


EXERCISES. 

Reduce  the  following  expressions  to  the  form 

X  —  {an  aggregate). 

1.  X  —  a  —  b.  Ans.  x  —  (a  +  b). 

2.  X  —  m  —  n. 

3.  a  -\-  X  —  3x  -{-  2y.  Ans.  x  —  {—  a  ■\-  3x  —  2y). 

4.  —  3b  -\-  X  +  2c  -It  hd. 

5.  2a;  —  2a  +  2b.  Ans.  x  —  {— x  +  2a  —  2b). 

6.  2x  -\-  a  —  b. 

7.  3x  —  2m  4-  2}i. 

8.  3x  -{-  ab  —  m  —  3ab  +  2m. 

9.  X  —  2m  —  {3a  —  2^).      A71S.  x  —  {2m  -f-  3a  —  2b). 

10.  a;  4-  3  —  (a  +  b). 

11.  X  -1^  a  —  {b  —  c)  -\-  {m  —  n). 

12.  X  —  {am  -\-  h)  —  {p  —  q)  —  {am  —  n). 

13.  X  —  {a  +  b)  —  {p  —  q)  —  {m  —  n). 


SUBTRACTION, 


37 


Compound  Parentheses. 

61«  When  parentheses  of  addition  or  subtraction  are  en- 
closed between  others,  they  may  be  separately  removed  by  the 
preceding  rules. 

Wo  may  either  begin  with  the  outer  ones  and  go  inward, 
or  begin  with  the  inner  ones  and  go  outward. 

It  is  common  to  begin  with  the  inner  ones. 

EXAMPLES. 

Clear  of  parentheses: 
,.    f-le-.\d-[c-{h-a)-\\\ 
Beginning  with  the  inner  parentheses,  the  expression  takes, 
in  succession,  the  foUowiijg  forms: 

f_le-\d-{c-h  +  a-]\^ 

=  f-[_e-\d-c-rh-a\\ 

=  f—[e  —  d-Vc  —  h-\-a\ 
=  f— e-\-d  —  c-\-h  —  a. 

2.     a;-[-(«  +  J)  +  (wj +«)-CT-y)]. 

Removing  the  inner  parentheses,  one  by  one,  we  have, 

x—[—a  —  h  +  m-\-n  —  x  +  y\ 
z=  X  +  a  -{-  b  —  m  —  n  -\-  X  —  y. 


EXERCISES. 

Remove  the  parentheses  in  the  following  expressions,  and 
combine  term^^.  containing  x  and  y,  us  in  §§  54  and  55. 
m  +  [-  {p  -  (?)  -h  («-*)  +  (-  c  +  r7)]. 
m  —  [—{a  —  b)  —  {])  +  q)  +  {fi  —  k)l 
7^a;  -  [{2ax  +  by)  -  {3ax  -  by)  +  (-  7ax  +  2by)]. 
a  —  {a—\a  —  [a  —  {a  —  a)]\]. 
p  ^  [a  —  b  —  {s  +  f  -\-  a)  -^  {—  m  —  71)1 
2ax  —  [Sax  —  by  —  {7ax  +  2by)  —  {6ax  —  3%)]. 
ax-^by-\  cz  +  [2ax—dcz  —  {2cz-\-bax)  —  {Uy—Zcz)], 
X  -  \  2x  -  y  -  [3x  -  2;/  -  {4x  -  dy)]  \. 
ax  —  bz—\  ax  +  bz  —  [ax  —  bz  —  {ax  +  bz)]  \. 
my  —  { a;  +  3?/  +  [2m^  —  3  (a;  —  ?/)  —  ^ab']  +  5 }. 


I. 

2. 

3- 

4- 

5- 
6. 

7- 
8. 

9- 

lo. 


\\ 


88 


ALOEliRAIC  OPERA  TIONS. 


11.  rt.r  4-  4cx  —  {mx  -\-  ex  —  y)  -f  [mx  —  {rx  4-  y)]. 

12.  'ittx  —  Ux  —  (  -  ^ay  —  3nz  +  'Sby)  —  Mz. 

13.  13flx  -f  2.r/^  -d-  l^ad  f  (ry  -f-  d)]  -  4rry. 

14.  w?  +  4.r  —  [—  4//  H-  2.«  +  (^//  —  ^)  -f  ;>]• 

15.  ^rtV^  -  3wi  -  [Z'v'i  -  Gil  +  {Vy  -  y  Vi/)]. 


eac 


-♦^■•- 


or 


CHAPTER    II. 

MULTIPLICATION. 

02.  Tlie  product  of  several  factors  can  always  l)e 
ex])r('sse<l  hy  writing  tlieiii  after  each  other,  and  enclos- 
ing tliose  wliicli  are  aggregates  within  parentheses. 

EXAMPLES. 

The  product  ot  a  +  b  hy  c  =  c  (a  +  b). 

f  4-  w  X  -\-  11 

The  product  of  '—~-  hy  x  —  y  =  {x  —  y)  '--,—  • 

The  product  ot  a  ■{-  b  hy  c  -{■  d  =  {c  -i-  d)  (a  +  b). 
Such  products  may  be  transformed  and  simplified  by  the 
operation  of  algebraic  multijilication. 

General  Laws  of  Multiplication. 

G.3.  Laio  of  Covfirmdatlon.  Multiplier  and  multi- 
plicand may  be  interchanged  without  altering  the 
product. 

This  law  is  proved  for  whole  numbers  in  the  following  way. 
Form  several  rows  of  quantities,  each  represented  by  the 
letter  «,  with  an  equal  number  in  each  row,  thus, 

a  a  a  a  a  a 

a  a  a  a  a  a 

a  a  a  a  a  a 

a  a  a  a  a  a 

a  a  a  a  a  a 


MULTIPLICATION. 


30 


y)J. 


ays  ha 

enclos- 
es. 


by  the 


multi- 
ig   the 


ng  way. 
by  tho 


Lot  in  bo  tho  miinbcr  of  rows,  aiul  n  tho  number  uf  a's  iu 
cacli  row.    Then,  counting  by  rows  there  will  be 

m  X  n  (luantiLies. 
Counting  by  columns,  tliere  will  be 

n  X  M  quantities. 
Therefore,  m  x  n  =  n  x  m, 

or  nm  =  mn. 

(»4.  Lrno  of  Association.  When  there  are  three 
factors,  //i,  n,  and  re, 

)n  {n(t)  —  (inn) a. 

Example.        3  x  (5  x  8)  =  .3  x  40  =  120.  ^ 

(3x5)x8  =  lo  +  8  =  120. 

Proof  for  Wlwh  Nnmbvn^.  If  a  in  the  al)ovc  sHiomo 
represents  u  number,  tho  Kum  of  each  row  will  be  lui.  Beeuuse 
there  arc  m  rows,  the  whole  sum  will  l)e  >if  (na). 

But  the  whole  number  of  r/'s  is  mn.    Therefore, 

711  (iia)  =  (inn)  a. 

Ciii,  Tlie  Dlslrlhut'n'e  Law.  Tlie  product  of  an  ag- 
gregate by  a  factor  is  equal  to  the  sum  of  the  products 
of  each  of  the  parts  which  form  the  aggregate,  by  the 
same  factor.     That  is, 

m  {p  -\-  q  +  r)  =  mp  +  mq  +  mr.  (1) 

Proof  for  Mliole  Numhcrs.  Let  us  write  each  of  the  quan- 
tities p,  7,  r,  etc.,  m  times  in  a  horizontal  line,  thus. 

P  +  P  -i-  P  -\-  etc.,  7n  times  ~  mp. 

q  -h  q  -}-  q  +  etc.,  vi  times  =  viq. 

r  +  r  +  r  +  etc.,  m  times  =  7nr. 

etc.  etc.           etc. 

If  we  add  up  each  vertical  column  on  the  left-hand  side, 
the  sum  of  each  will  be  ^^  +  (/  +  ^*  +  etc.,  the  columns  being 
all  alike. 

Therefore  the  sum  of  the  in  columns,  or  of  all  the  quanti- 
ties, will  be 

m{p  ■}•  q  -{-  r,  etc.). 


II 


ll 


(S- 


40 


ALGEBRAIC  OPERATIONS. 


The  first  horizontal  line  of  /?'s  being  mp,  the  second  mq^ 
etc.,  the  sum  of  the  right-hand  column  will  be 

mp  +  mq  +  mr,  etc. 

Since  these  two  expressions  are  the  sums  of  the  same  quan- 
tities, they  are  equal,  as  asserted  in  the  equation  (1). 


3Iultiplicatio<.  of  Positive  Monomials. 

G6.  Rule  of  Exponents.     Let  us  form  the  product 

a;"»  X  x\ 

I^y  §  37,     x""  means  xzx,  etc.,  taken  m  times  as  factor. 
x"  means  xxx,  etc.,  taken  n  times  as  factor. 

The  product  is  xx.xx,  etc.,  taken  {in-\-n)  times  as  factor. 

Therefore,  af*  x  a:"  =  ic'"^". 

Hence, 

Theorem.  The  exponent  of  the  product  of  like  sym- 
bols is  the  sum  of  the  exponents  of  the  factors. 

67.  As  a  result  of  the  laws  of  commutation  and 
association,  the  fr*,'ioi.>  jf  a  product  may  be  arranged 
and  multiplied  \is  ?noh  order  as  will  give  the  product 
the  simplest  form. 

08.  Any  product  rf  monon  ials  may  be  formed  by 
combining  these  principles. 

Example.    Multiply  h  11111^01^1/  by  '^hnx^y. 

By  the  rules  of  algebraic  language,  the  product  may  be  put 

into  the  form 

bmn^j^i/lhnx^y. 

By  interchanging  the  factors  so  as  to  bring  identical  sym- 
bols together, 

h-1  h  m  n^ nx? a^  1/ y. 

Multiplying  the  numerical  factors  and  adding  the  exponents, 

the  product  becomes 

35  ft  m  n^  ^  y\ 


M  UL  TIP  Lie  A  TION. 


U 


nd  mq, 


le  quan- 


S. 


tor. 
tor. 

factor. 


^e  syni- 

on  and 
ranged 
)roduct 


aed  by 


y  be  put 


?al  sym- 


ponents, 


69.  We  thus  derive  the  following 

Rule.  Midtiphj  the  ninncrlcnl  coefficients  of  the 
factors,  affix  all  the  literal  parts  of  the  factors,  and  give 
to  each  the  sum  of  its  exponents  in  the  separate  factors. 


A71S.  i^y\ 


3- 

5- 
7- 
9- 


Multiply  hni^ij  by  "im^x. 
Multi})ly  'Zam  by  'ima. 
Multii)ly  'ixyz  by  'ixyz. 
Multiply  3rtZ'2.c3  ^y  'da%\x. 


13.  Multijily -7  M^/;  by  4wi^*. 


EXERCISES 

1.  Multiply  xy  by  x'^y. 

2.  Multiply  3rt.T  by  2ahx^. 
4.  Multi})ly  "Z\my  by  'Zahn. 
6.  Multiply  bx''yh  by  x^yH. 
8.  Multij)ly  '^abm  by  2mba. 

10.  Multiply  2-(})np(jr  by  2-Gpqrs. 

11.  Multiply  12«.T^  by  12a-^z. 

3  2 

12.  Multiply  -mV  by  ^.m^y'. 

7 
14.  Multiply  -  aJcc?  by  4f?r/^. 

70.  When  we  have  to  find  the  product  of  three  or  more 
quantities,  we  multiply  two  of  them,  then  that  product  by  the 
third,  that  product  again  by  the  fourth,  and  so  on. 

Ex.     2ah  X  2«2J  x  3«J2  x  Umxy  =  3(ja*l/^mxy. 

Exercises.     Multiply 

1 5 .     mx  X  7)1  y  X  mz.  16.     axx  hx  x  ex  x  dx. 

dahn  x  4:b^n  X  m7i.  18.     abx  2bc  x  lea. 

dmn^  X  5)ip^  X  9pm?'. 

abxacx ad x am3  xyx 2yz x zx. 

amx  X  a?ix  x  amxy  x  anxy  x  amxyz. 


17- 
19. 
20. 
21. 
22. 

23- 


a^x  X  a^y  x  ax^  x  ay'^  x  a\i^  x  a^y^  x  xhf. 
2am  X  'Sail  x  a^  x  m^  x  4:mx  x  2nx. 


Rule  of  Signs  in  Multiplication. 

71.  It  was  shown  in  §  25  that  a  product  of  two  factors  is 
)>()sitive  when  the  factors  have  like  signs,  and  negative  when 
they  have  unlike  signs.     Hence  the  rule  of  signs, 

+   X   +     makes     +, 

+   X  - 

-  X   + 

-  X  -        "         +. 


it 


« 


i!! 


I 


ALGEBRAIC  OPERATIONS. 


Examples.    The 

quantity  a 

Multiplied  by 

3 

2 

makes 

+  3«. 
+  205. 

(( 

a 

1 

a 

+    a. 

u 

te 

0 

a 

0. 

a 

a    _ 

-1 

u 

—   a. 

a 

«    _ 

-2 

a 

—  2a. 

The  quantity  —  ( 

% 

Multiplied  by 

3 

2 

makes 

—  3fl5. 

—  2a. 

a 

a 

1 

a 

—   a. 

a 

li 

0 

a 

0. 

cc 

a    _ 

-1 

a 

+    a. 

a 


a 


a 


+  2a. 


73.  Geometrical  lUufttrafion  of  the  Rule  ofSigtis.  Suppose 
the  quantity  a  to  represent  a  length  of  one  centimetre  from 
the  zero  point  toward  the  right  on  the  scale  of  §  11. 

Then  we  shall  have 

a  =  this  line    |  | 

The  product  of  the  line  by  the  factors  from  -1-3  to  —3 
will  be 


a  X  S, 

a  X  2, 

a  X  1, 

a  X  0, 

a  X  —1, 

a  X  —2f 

a  X  —3,      I 

We  shall  also  have 


0 


a 

0 


I 

0 


—  a  =  this  line    f 


MULTIPLICATION. 


43 


The  products  by  the  same  factors  will  be 

0 

- «  X  3,      I  ,  ,  1 


—  a  X  2, 

—  rt  X  1, 

—  «  X  0, 

—  «  X  —  1, 

—  ax  —  3, 

—  rt  X  —  3, 


0 

0 


r 

0 


These  results  are  embodied  in  the  following  two  theorems  : 

1.  Multiplying  a  niao;iiitude  by  a  negative  factor, 
multiplies  it  by  the  factor  and  turns  it  in  the  opposite 
direction. 

2.  Multiplying  by  —1  turns  it  in  the  opposite  direc- 
tion without  altering  its  length.  . 

Note.  When  more  than  two  factors  enter  a  product,  the 
sign  may  be  determined  by  the  theorem,  §  :^G. 


I 

3 

4 

5 
6 

7 
8 

9 
lo 

II 

12 

13 
14 
15 


EXERCISES, 

am  X  ah  X  ac  X  ad.  2.     ax  x  - 

X  X  — ax  X  — ahx  x  — ahcx. 
^ax  X  —'la%^  X  —  ha^mx, 

—  7i)i'^i/  X  —''^ahf'  X  5ax. 
—2uzn  X  — 5mV'  X  — ?i^i/z  —  x!^, 
2m  X  n  X  —a  x  —2b. 

—Sax  X  —2 /cm  x  —7x  x  —^bmx, 

—  uy  X  (ly  X  —2yx  Sbm. 
xy  X  2/y2  X  y^x  x  2ayxK 

•5?/2  X  — -V/?/  X   —2z'^  X  —ax^z. 
bax  X  anx  x  3;;  x  b^xy. 

—  ^bz  X  —xz  X  —yz  X  agz. 
2c^ii  X  2/h  X  —z^x  —hgz\ 
—(Px  X  3^  X  cb^  X  ay. 


•hx  X  ex  X  clx. 


i) 


44 


ALGEBRAIC  OPERATIONS. 


1 6. 

17. 
18. 

19. 
20. 
21. 
22. 

23. 

24. 

25- 

26. 

27. 
28. 
29. 

30. 


— 2e  X  — 2y  X  rt  X  5a;. 

—4a.?;  X  3«y  x  —"^a^y  x  —  a;^. 

rt'^:c  X  — aif  X  ax^  x  —x-y. 

ax^  X  — ^'^  X  —1  X  ^ax  x  —  a^ 

?m2.c  X  — n^x  X  — wj?i^  X  mx  x  —  7?i*. 

— «5a;  X  —ay^x  ax  x  «'^.^'^. 

px^  X  (72/^  X  .ry  X  —ax. 

abc  X  — £?^  X  aa;2  X  —  1  X  Sax. 


-jttx  X  3ca;  x 
4 


^  wo;  X  —  4^/^  X  Gm. 


—Qmx  X  —2h^x  X  7;ac  X 

0 


rWi'' 


—a  X  Jc  X  — 1  X  T  X  3a2  X  ixy  x  y. 

—1  X  «a;  X  a^x  x  a^x^  x  bx  x  d. 
—anx  2cfm2  x  —d)nn  x  ^n^y  x  —m, 
—mx  X  nx  X  —mn  x  —xy  x  —  1. 

—2px  X  —dqx  X  ^mH  X  -zy^  X  — 1. 


Products  of  Polynomials  by  Monomials. 

73.  The  rule  for  multiplying  a  polynomial  is  given  by  the 
distributive  law  (§  65). 

Rule.  Multiply  each  term  of  the  polynomial  by  the 
monomial,  and  take  the  algebraic  sum  of  the  products. 

Exercises.    Multiply 

1.  dix^  —  ^xy  —  5y2  by  —  Aax. 

A71S.  —  12ax^  4-  IGax^y  +  20axi/\ 

2.  3.?;2  —  xy-\-  y^  by  3.c. 

3.  x^  ■}-  xy  4-  ?/2  by  Sx.  4.     ax  -\-  hy  -{-  cz  by  axyz. 
5.     Saofi—bmf—^  by  ^ahx.     6.     4:mp  —  Gnq  by  —  'dmq, 
7.     ba?y^  —  7a^y^  —  7a*y  by  Sab. 

74.  The  products  of  aggregates  by  factors  are  formed 
in  the  same  way,  the  i)arentheses  being  removed,  and 
each  term  of  the  aggregate  multiplied  by  the  factor. 


MUL  TIP  Lie  A  TION. 


45 


Example.    Clear  the  following  expression  of  parentheses: 

am  {a  —  h-{-c)  —])  [a  —  {h  —  k)  —  m  {a  —  h)]. 

By  the  rule  of  §  73,  the  first  term  will  be  reduced  to 

ahn  —  amb  +  amc.  (1) 

The  aggregate  of  the  second  term  within  the  large  paren- 
theses will  be 

a  —  li-\-k  —  m  {a  —  b) 

z=a  —  h-\-k  —  ma-\-  mh,  (3) 

because,  by  the  rule  of  signs  in  multiplication, 

—  m  {a  —  h)  =  —m  x  a  —  m  x  —b  =  —  ma  -\-  ml. 

Multiplying  the  sum  (2)  by  —  p  and  adding  it  to  (1),  we 
have  for  the  result  required: 

a^m  —  ami)  +  amc  —  pa  -\-2)h  —  ph  +  pma  —pml. 


EXERCISES. 

Clear  the  following  expressions  of  parentheses  : 

1.  p  {a  -\-  m  —  p)  -{-  q{b  —  c)  —  r{b  -^  c). 

2.  (m  —  an)  x  —  {?n  +  a7i)  y  +  {an  —  m)  z. 

3.  a{x  —  ij)c  —  b{x  —  y)d+f{x-{-y)  cd. 

Here  note  that  the  coefficient  of  x  —  y  in  the  first  term  is  ae. 


4 

5 
6 

7 
8 

9 
10 


am  [x  —  a{b  —  c)]  —  b7i  [ax  +  i  (c  -f  d)]. 

p  [—  a  {m-\-?i)-\-b  (m—n)]  —  q[b  {m—n)—a{m-\-n)]. 

3.r  {2q  —  7ic)  -r  2y  (5.r  —  3c)—z  (2m  +  7m)- 

am  [m  {a  —  b)c  —  Sh  {2k  —  4rf)  +  4w]. 

2pq  [3a  —  bb  —  6c  —  pq  {2m  —  3n)]. 

In  [_  7rt  _  7^»  (rt  —  c)  —  (3  —  a  -  b)\ 

P{q  —  r)  +  q{r  —  p)  +  r{p  —  q). 


75.  The  reverse  operation,  of  summing  several  terms  into 
one  or  more  aggregates,  each  multiplied  by  a  factor,  is  of  fre- 
quent application.    Thus,  in  §  65,  having  given 

mp  +  7nq  -f  mVf 

we  express  the  sum  in  the  form 

m  {p  +  q  +  r). 


If 

ilii 


ill  I 


46 


ALGEBRAIC   OPERATIONS. 


The  rule  for  the  operation  is 

//  the  sum  of  several  terms  having  n  cmnmnn  facinr 
is  io  he  formed,  the  coejfieients  of  this  foetor  may  he 
added,  and  their  aggregate  he  midti/died'  hij  the  factor. 

Note.     This  (jperutiou  is,  in  piiuciplc,  idonticul  with  that  of  g  55, 

EXAMPLES. 

dbx  —  hex  —  (uhj  +  3di/—  3bx  -\-  4:adi/  +  my—amy—'icmx  -\-  hmx. 

Collecting  the  coefficients  of  x  unci  y  as  directed,  we  have 
{(d)  —  be  —  3b  —  dcm  +  bm)  x  +  {—nd-\-3d-\-4ad+m—a7n)  y. 

Applying  the  same  rule  to  the  terms  within  the  parentheses, 

we  lind 

ah  —  be  —  ?,h  =  b{a  —  c  —  3). 

—  3em  +  bm  =  m  {b  —  3c). 

—  ad-{-  3d  +  4.ad  =  3ad  +  3d 

=  {3a  +  3)  d 

=  3{a  +  1)  d. 

m  —  am  =  m.  (1  —  a). 

Substituting  these  expressions,    the    reduced    expression 
becomes 
[^,  (a  _  c  —  3)  +  m  (b  —  3c)]  x  i- [3  {a -{- 1)  d  -\-  m.  (1  —  «)] ;/. 

The  student  should  now  be  able  to  reverse  the  process,  and 
reduce  this  last  expression  to  its  original  form  by  the  method 

of  §  74. 

EXERCISES. 

In  the  following  exercises,  the  coefficients  of  y,  2,  and 
their  products  are  to  be  aggregated,  so  that  the  results  shall 
be  expressed  as  entire  functions  of  x,  y,  and  z,  as  in  §  55. 

1.  ax  -\-  bx  —  3ax  +  3bx  ■\-  Qx  —  7x. 

Ans.  {— 'Ha -\- '\:b  —  1)  X. 

2.  my  +  py  —  my  —  2py  —  3r/y. 

3.  mx  —  ny  -\-  jtx  —  <jy  -\-  rx  —  sy. 

Ans.  {in  -\-  ]>  -\-  r)  x  —  {n  -\-  g  -\-  s)  y. 

4.  3az  —  y  —  2az  -[■  z  —  az  -{-  y. 


MUL  TIP  Lie  A  TION. 


47 


Zxy. 


5.  ahxy  —  hcxji  -\-  hdxy. 

6.  'dijab.iij  —  "Z^x  —  ax  - 

7.  ay  —  by  —  may  —  nby  +  ^•^« 

8.  amy  —  bmij  -\-  any  —  bay. 

9.  pi'z  —  2qrz  —  4pj)Z  +  8(//iz. 
10.  c?ix  4-  i;w;  —  «v>/^  —  'Zbny. 

10.  All  entire  function  of  two  quantities  can  be  regarded 
as  an  entire  function  of  either  of  them  (§§  49,  50),  and  wlien 
expressed  as  a  function  of  one  may  be  transformed  into  a  func- 
tion of  the  other. 

Example.    Tlie  expression 

{2a  +  3)  a.-3  —  (-l«2  _  2a)  .t^  +  {a^  —  2a -{■  1)  x  —  a^ 

lias  the  form  of  an  entire  function  of  x.     It  is  required  to 
express  it  as  an  entire  function  of  a. 
Clearing  of  })arontlieses,  it  becomes 

2ax^  -\-  3.j;3  —  4^2:^2  4-  ^r/.^a  4-  a^x  —  2ax  +  x  —  a^ 

Now,  collecting  the  coefficients  of  a^,  a^,  etc.,  separately,  it 
becomes 

( _  4.7;2  ^  a;  —  1 )  a^  -\-  {2x^  4-  2x^  —  2x)  a  +  3x^  +  x, 

which  is  the  required  form. 


EXERCISES. 

Express  the  following  as  entire  functions  of  y : 

1.  (3,y2_4^/).^3_,_(2/3_o^2^i)  ^^ ^ {2y^ +  by^—':!)x—y^-G. 

2.  (y*  —  if)  x^  4-  (?/3  —  y)x  4-  y^  ~  1. 

3.  if  -  2//)  .^•«  +'(//  -  2//2)  .r^  4-  (/  -  2//)  X  4-  y'  -  2. 

4.  (v/5  +  3/0  -^-^  +  {f  +  ^f)  ^^  +  (^'  +  %)  •^■'^  4-  (i/2  4-  3)  2:. 

3Iultii)licjition  of  Polynomials  by  Polynomials. 

et  us  consider  the  product 


i  i, 


{a  4-  b)  {p  +  q  +  ?'). 


This  is  of  the  same  form  as  equation  (1)  of  §  G5,  {a  4-  b) 
taking  the  place  of  m.    Therefore  the  product  just  written  is 


equ 


d  to 


{a  +  b)p  +  {a  +  b)q  4-  {a-[-b)r. 


.<! 


48 


ALGEBRAIC  OPERATIONS. 


But  {a  +  h)2^  =  ftp  +  b^h 

(a  -\-  b)  q  =1  aq  -\-  bq. 

{a  -\-  b)  r  =^  ar  +  br. 
Therefore  the  product  is 

ap  4-  bp  -\-  aq  -\-  bq  +  ar  +  Jr. 

It  Avould  have  been  still  sliorter  to  first  clear  the  paren- 
theses from  (rt  4-  b),  putting  the  product  into  the  form 

a{P  +  q  +  r)  ■\-b{p  +  q  +  r). 

Clearing  the  parentheses  again,  we  should  get  the  same 
result  as  before. 

"VVe  have  therefore  the  following  rule  for  multiplying  aggre- 
gates: 

78.  Rule.  Multiply  each  term  of  the  niultipUeajid 
hy  each  term  of  the  midtlplier,  and  add  the  coefficients 
with  their  proper  algebraic  signs. 


EXERCISES. 


{a  4-  b)  (2r.  -  bn^  —  2bti^). 
{a  —  b)  (3wi  +  2n  —  habmn). 
{m^  —  n^)  {2m)i  +  jnn  +  q?i). 
(;j2  4-  ^2  +  7-2)  {pq  4-  qr  4-  rp). 
(2a  —  3i)  {2a  f  2b). 
{mx  —  ny)  {mx  4-  ny). 

79.   It  is  frequently  necessary  to  multiply  polynomials 
containing  powers  of  the  same  letter.     In  this  case  the  begin- 
ner may  find  it  easier  to  arrange  multiplicand,  multiplier,  and 
product  under  each  other,  as  in  arithmetical  multiplication. 
Ex.  I.    Multiply  W  —  Qx^  +  bx  —  i  by  3a;2  _  4^  _  5. 
The  first  line  under  work. 

7.^-3_Ga:2  4-5a;— 4 
U^—ix  —5 


tlie  multiplier  contains 
the  products  of  the  sev- 
eral terms  of  the  multipli- 
cand by  3a;''.  The  second 
contains  the  products  by 
— 4j*,  and  the  third  by  —5. 
Like  terms  are  placed 
under  each  other  to  facil- 
itate the  addition. 


21a;5_iaT4  4-  Ibx^—I2xi 

— 28.c4  4-24a;3_20.t;2  4-10a: 
— 35a:3  4-  ^Qx^—2bx-\-20 

2\7^—^(jx^+  4a;3—  2x^—  ^x+20 


MULTIPLTCATION. 


49 


same 


5. 


Ex.  2.    Multiply  7)1  +  nx  +  px"^  by  «  —  bx. 
m  +  nx  +  2^^^ 


a 


bx 


am  -f-  anx  -{•  apu^ 
—  bmx  —  biix^ 


bpa^ 


am  +  {an  —  bm)  x  +  {ap  —  bn)  x^  —  bpT? 

In  the  folluwin<T  cxeroiKcs  arrange  the  terms  according  to 
the  powers  and  i)roduct8  of  the  leading  letters,  «,  bj  x,  y,  or  z. 

Multiply 

3a^  +  5a  +  7  by  'Za^  —  3«  +  4. 
a^  4-  ab  -^  b"^  hy  a  —  b. 


I. 

2. 

3- 

4. 

5- 
6. 

7. 
8. 

9- 
lo. 

II. 

12. 

13- 

14. 

15. 
16. 

17. 

18. 

19. 

20. 

21. 
22. 

23- 

24. 

25- 

26. 


a^  +  a^  +  r/w^  4-  x^  by  a  —  a;. 

«3  —  a^  ^  a  —  1  by  a^  —  a  +  1. 

ar*  +  aa^  -f  «^.c2  ^  ^3^  _|_  ^^4  |jy  ^  —  a. 

a  +  bz  +  cz^  4-  cl^  by  ?«.  —  W2;  -j-  2JzK 

3a8  f  5a  +  7  by  2^2  +  3a  —  4. 

a^  —  ab  +  b^  by  a  -\-  b. 

a^  +  r/S^  -\-  ax"^  +  x^  hj  a  —  X. 

«2  —  a^  +  a  —  1  by  a^  _|_  ^  —  1. 

X*  +  ax-^  +  a^x^  +  rt%  +  a*  by  a;  +  a. 

1  +  bz  -\-  cz^  -\-  dz^  by  m  +  7iz  —pz\ 

(a  4-  bx)  {m  -\-  nx). 

{a  +  bx  4-  f.T^)  (m  4-  'y«a;  4- j^a:^). 

(/  -  3^  4-  2)  if  ~  2). 

(y3  4.  2/2  +  y  _!_  1)  (^2  ^_  y  ^  1) 

(y3  _  5>y.  4-  3//  -  4)  if  4-  2/  4-  3^  4-  4). 
da-"'x  —  3a2_y  -(-  2^2"  by  a'"  —  a" 

a^  4.  Qab  -\-  r.b  hj  a  —  ;^b. 

{a  4-  i)  4-  («  —  />)  by  {a  i-  b)  —  {a  —  b). 
(fi  —  p^[a  —  b)  by  a2  4-  Z»«  4-  (a  4-  ^'). 
a  -{■  b  -}-  c  hj  a  —  b  -\-  c. 
r<2  4-  ^2  _  (3^<2  ^  ^,2)  by  2a  4-  2J  —  2  (a  —  b), 

2(a  —  b)+x  —  y  hy  a  -\- b  —  {x  +  y). 
ax'"-  4-  bx"  —  abx  by  a^  4-  Z'a;^. 
a"'  —  J"  by  a'"  4-  b\ 


ilil 


r 


60 


27. 


2 


ALOEBRAfC   OPERATJONS. 

ir^x^y  +  3.r//2  —  Uf  by  —  Tx/y. 

1 


28.     .,.^3  4-  3rt.r  —  .  rt'  l)y  2:1-3  _  ^^-^  _    ^i. 

NoTi:.  A^T^gro^'iites  oiitoring  into  either  factor  should  bo 
siinplilied  before  inuUiplying. 

Special  Forms  of  Multiplication. 

80.  1.  To  find  the  square  of  a  binomial,  aa  a  +  b.  "Wo 
multiply  a  -\-  b  by  a  -{■  b. 

a  (a  -t-  Z>)  =  a'  +  a5 

b  {a  -\-b)  =  ab  +  ^ 

(a  -^b){a-^b)  =  «2  +  ''iftb  +  b^ 

Hence,  {a  +  by  =  a^  +  2ab  ^-i-  U^  (1) 

2.  We  find,  in  the  same  way, 

(rt  -  by  =  rt2  _  2ab  +  b\  (2) 

These  forms  may  be  expressed  in  words  thus: 

Tlieorem.  The  square  of  a  binomial  is  equal  to  the 
sum  of  the  squares  of  its  two  terms,  jjIus  or  minus  twice 
their  product. 

3.  To  find  the  product  of  «  +  J  by  «  —  Zi. 

a{a  ■\-b)  =■  c^  ^  ab 
—  b{a  +  ^)  -      —ah-W 
Adding,    {a  +  b)  [a-b)  =a^-  b\  (3) 

That  is : 

Theorem.  The  product  of  the  sum  and  difference  of 
two  numbers  is  equal  to  the  difference  of  their  squares. 

The  fonns  (1),  (2),  and  (3)  should  be  meinorized  by  the  student,  owing 
to  their  constant  occurrence. 

When  J  =  1,  the  form  (3)  becomes 

{a  +  1)  (rt  _  1)  =  a2  -  1. 

The  student  should  test  these  formulae  by  examples  like 
the  following : 

(9  +  4)2  =  92  +  2.9-4  +  42  =  81  +  72  +  IG  =  109. 
(9  _  4)2  =  92  —  2-9.4  +  42  =  81  -  72  +  IG  =  25. 


n 


it 


a 


nc. 

wli 
thi, 

wli 


MULTirUCATION. 


61 


Wo 


(1) 
(2) 


(3) 


A 


(9  ^  4)  (9  _  4)  =  92  _  4.  =  r.5. 

Prove  tlicse  three  C([uatioiis  by  computing  the  left-ljuiul 
mcniMer  directly. 

EXERCISES. 

"Write  on  sight  the  values  of 

I.     {m  +  )tn)\  2.     (m  —  2n)'. 

3.    {:Mi-^zbY.  4.    (-i.^-.^y)2. 

5.     {-Ix  +  y)  {'Ix  -  y),  6.     (32:  +  1)  (32- -  1). 

7.     (4x2  +  1)  (■ia:«  -  1).  8.     (oar^  -  3)  {bu^  -f  :3). 

81,  Because  the  product  of  two  negative  factors  is  positive, 
it  follows  that  the  square  of  a  negative  quantity  is  i)o.sitive. 

Examples.      (—  af  =  ««  =  (+  ay. 

(i  _  af  =  a^  _  ^lab  +  i^  =  (a  -  h)\ 

Hence, 

TliG  cxprcsfiinii  a'  —  2«5  +  V^  is  the  S(/uaj'e  huth  of 
h  ami  of  b  —  a. 


a 


—  a  X  a  ^ 


a\ 


83.  We  have 
Hence, 

Tlie  proiliict  of  equal  factors  with  opposite  si^iis  is  a 
negative  square. 

Example.      —{n  —  l)  (a  _  Z<)  =  _  «»  +  2ab  —  b^ 

which  is  the  negative  of  (2).     Because  —  {a  —  b)  =  b  —  a, 
this  equation  nuiy  be  written  in  the  form, 

(5  _  a)  {a  -b)  =  —a^  +  'Zab-  1?, 

which  is  readily  obtained  by  direct  multiplication. 

EXERCISES. 

Write  on  sight  the  values  of 

I.     —  (rt  +  5)  X  —  (a  4-  b). 

2-     (-^  -  y)  (!/  —  -c)-  3-     (^  +  y)  (-a:  —  y). 

4.     {2a  -  3b)  {3b  -  2a).  5.     (3*  -  "ia)  {-  3b  +  2a). 

6.     {am  —  bn)  {bn  —  am).         7.     {.ry  _  2)  (2  —  xy). 


M\ 


52 


A LflKHn.  1 W   OPKllA  riONS. 


V 


CHAPTER    III. 
DIVISION. 

8,'J.  Tlic  problem  of  jil;rol)rjiic  division  is  to  find  sucli  un 
expression  tliut,  when  miillipliod  by  the  divisor,  the  product 
sliull  l)C  the  dividend. 

Tiiis  expression  is  called  the  quotient. 

In  Algcbrii,  the  (piotient  of  two  quantities  niuy  always  bo 
indicated  by  a  fraction,  of  which  the  numerator  is  the  divi- 
dend and  the  denominator  the  divisor. 

Sometimes  the  numerator  cannot  be  exactly  divided  by  the 
denominator.  The  expression  must  then  be  treated  us  a  frac- 
tion, l)y  methods  to  be  ex])lained  in  the  next  chapter. 

Sometimes  the  divisor  Avill  exactly  divide  the  dividend. 
Such  cases  form  the  subject  of  llie  present  cha[)ter. 

Division  of  3[oiioiiiial.s  by  3IoiioiniiilM. 

84.  Ill  order  tliat  a  divideiul  may  bo  exactly  divisi- 
ble by  a  divisor,  it  is  necessary  that  it  shall  contain  the 
divisor  as  a  factor. 

Ex.  I.     15  is  exactly  divisible  by  3,  because  3' 5  =  15. 
2.  The  product  ab-c  is  exactly  divisible  by  ac,  because  ac  is 
a  factor  of  it. 

To  divide  one  expression  by  another  which  is  an  exact 
divisor  of  it: 

Rule.  licinoie  from  the  (livid end  those  factors  the 
product  of  irhich  is  eqival  to  the  divisor.  The  reniaiii- 
ing  factors  ivill  he  the  c/uotieiit. 

85.  Rule  of  Exponents.  If  ])oth  dividend  and  divisor 
contain  the  same  symbol,  with  different  exponents,  say  m  and 
w,  then,  because  tlie  dividend  contains  this  symbol  m  times  as 
a  factor,  and  the  divisor  n  times,  the  quotient  will  contain  it 
?«  —  71  times.     Hence, 


ti 


nil 


tic 


tha 
bra 


DIVISION. 


68 


exact 


\ 


Til  dividing,  exponents  of  like  symbols  arc  to  he  sub' 
tracted. 

EXERCISES. 

1.  Divide  207?/  by  'lij.  Ans.  13.r. 

2.  J)ivi(lo  -lidHtc  by  ''ibc, 

3.  Divide  r^  by  x"^.  Ans.  x, 

4.  Divide  \^<d^  by  Gat.  Ans.  'Ml. 

5.  Divide  \hd^m  by  3^.  Ans,  ham, 

6.  Divide  XWn^  hy  Ham. 

7.  Divide  Hki^tn*  by  8^/3,;/«.  8.   Divide  :H',.nfz^  by  G.n/^. 
9.  Divide  -lOaW  by  lOalt,?^.  10.  Divide  [ibul/^  by  ;«i^. 

Kiile  of  Slffiis  ill  Division. 

86.  Tlio  rule  of  signs  in  division  corresponds  to  that  in 
multiplication,  namely: 

If  dividend  and  divisor  have  the  same  si£n,  the  quo- 
ticnt  is  positive. 

If  they  have  opposite  si^ns,  the  cpcoticnt  is  negative. 

Proof. 

•\-m.v  -T-  i  +  ni)  =  -\-x,  because  -\-x  x  (  +  w)  =  -\-7nx. 

-\-mx  -r-  {  —  m)  =  —X,       "  — ^'  X  (— wj)  =  -{-mx. 

—mx  -h  {  +  ni)  =  —X,       "  —X  X  (  +  w)  =  —?nx. 

—mx  -7-  {—m)  =  +.r,       "  +.t  x  {  —  m)  =  —mx. 

Tlic  condition  to  be  fulfdlcd  in  all  four  of  these  cases  is 
that  the  product,  quotient  x  divisor,  shall  have  the  same  alge- 
braic sign  as  the  dividend. 

EXERCISES. 


Divide 

I.      +  a  by   +  a. 

Ans.  +  1. 

2.      -\-  a  by  —  a. 

Ans.  —  1. 

3.     —  a  by   +  a. 

Ans.  —  1. 

4.     —  a  l)y  —  a. 

Ans.   +  1. 

5.     —  '.y^ahnx  by  Wax. 

A 

ns.  —  ',}am. 

6.      —  24.r2y/z  by  XZryz. 

Ans.  —  2x. 

7.     'Ziani-x'"  by  —  ;a//<x". 

A 

ns. 

—  '3mx"'-". 

m 


iiii 


t; 


04 


ALGEBRAIC  OPERATIONS. 


8.  —  ISa^y/"  by  —  Ca"/?.  Ans,  ^a^'^})^-^. 

9.  —  IGa^x^y'^  by  Aa.r^i/\ 

10.  14i*//  by  —  :^//?. 

11.  —  12^^*X:'»  by  —  4Z'"M'». 

12.  12  {a  -  bye*  by  3  {'i  —  If  c.       Ans.  4  {a  —  V)  (^, 

13.  42  {x  —  ?/)'»  by  —  7  {.c  —  yyK 

14.  —  44rt«  {x  —  t/Y  by  11«'  (.<;  —  ;/)<. 

15.  —  45^  {a  —  by  by  yi«  («  —  i)^ 

16.  —  48  {m  -f  w)''  kv  —  8  (wi  4-  «)^. 

17.  G4  {a  4-  ^)''  {x  —  y)"'  by  4  («  +  Z/)  (a:  —  y). 


Division  of  Poljiioniials  by  Moiioiiiial.s. 

87.  By  the  distribntive  law  in  multiplication,  whatever 
quantities  the  symbols  m,  a,  b,  c,  etc.,  may  represent,  we  have: 

(rt  +  5  -}-  c  +  etc.)  X  m  =  ma  +  mb  +  mc  +  etc. 
Tlicrcfore,  by  the  condition  of  division, 

{iim  +  mb  +  mc  +  etc.)  -7-  m  =z  a  ■{■  b  -\-  c  +  etc. 
We  therefore  conclude, 

1.  In  order  that  a  polynomial  may  be  exactly  divisi- 
ble by  a  monomial,  each  of  its  terms  must  be  so 
divisible. 

2.  The  quotient  will  be  the  algebraic  sum  of  the 
separate  quotients  found  by  dividing  the  different  terms 
of  the  polynomial. 

EXERCISES. 

Divide 

1.  2«2  +  6a^x  —  8rt5j4  |,y  2^2.     A71S.  1  -f  3rta;  —  ^a^x\ 

2.  Gm^n  —  12m'/i*  —  li<nui^  by  Ctinn. 

3.  Sfi^ffi  —  lOrr^b*  +  Sr/5^/3  by  Aa^bK 

4.  4r//^  —  Sx^y^  -f  Ax^y  by  —  4.ry. 

5.  Viabx—  2-iabx^  by  —  12^7^?-. 

6.  21^/«2/-^  —  Ua-m*x^  -f  2Sa^m^x^  by  —  7a7n3f^. 

7.  T2rif3^  _|_  04^^  _|_  4fiffjfS  ]^y  2Aax. 

8.  a{b  —  c)  J^  b{c  —  a)  +  c  {a  —  b)  -{-  abc  by  abc. 

9.  27  (^^  -  ^)'  -  18  (a  -  //)3  +  0  (a  -  i)8  by  9  {a  -  b). 
10.     «»»  ((f  —  i)«  —  a'*  {((  —  b)""  by  «"  (a  —  i)". 


wli 
Th 


DIVISION. 


m 


I 

I 


II.     {a  +  hy  (a  -  by  +  {(i-[-by  {a-hy  by  {n  +  h)  {a-h). 

12.    10  {x  +  y)'^{x  -  yY  -  5 1^  +  yY  {■(■  -  yf 

by  5  (a;  +  2/)  (a;  -  y). 
13.     («  +  J)  (rt  -  h)  by  a2  _  ^^a. 

Factors  and  Multiples, 

88.  As  in  Arithmetic  some  numbers  are  composite  inid 
others  prime,  so  in  Algebra  some  expressions  admit  of  bciiii;- 
divided  into  algebraic  factors,  while  others  do  not.  I'iio  lalU-r 
are  by  analogy  called  Prime  and  the  former  Composite. 

A  single  symbol,  as  a  or  x,  is  necessarily  prime. 

A  ]»roduct  of  several  symbols  is  of  course  composite,  and 
can  be  divided  into  factors  at  sight. 

A  binomial  or  polynomial  is  sometimes  prime  and  some- 
times composite,  but  no  universal  rule  can  be  given  for  dis- 
tinguishing the  two  cases. 

89.  When  the  same  symbol  or  expression  is  a  factor  of  all 
the  terms  of  a  polynomial,  the  latter  is  divisible  by  it. 

EXAMPLES. 

1.  ax  +  ahx^  +  a^csi^  =z  a  {x  -\-  hx^  -\-  ac.c'). 

2.  aWx  +  rr^Z»V  —  aW-x  {h  +  ax). 

3.  «'^'*  +  a"x"  —  rt'i  (rt"  +  .^•«). 


EXERCISES. 


Factor 

I.     ax^  -f-  a'^x. 

3.     rt2«//«  +  a'^b^. 

5.       ««  Z>^»  C^n  +  ft""  ^»3n  en  _^  ^3n  ^n  (,-in^ 


2.     a%hy  -\-  aV)c^y  +  ab^cy. 

4.      ft3«  ^n  _  ((in  ^5n  _|_  ^n  ^^Qn^ 


90.  There  are  certain  forms  of  composite  expressions 
which  should  be  memorized,  so  as  to  be  easily  recognized. 
The  following  are  the  inverse  of  those  derived  in  §  80. 

1.  «2  4-  Ub  +  b"^  =  {a  -{■  b)\ 

2.  «'i  —  2ftZ>  +  /y2  —  [a  —  by 

3.  ft2  -  b'i  =  (ft  -H  b)  (ft  -  b). 

The  form  (3)  can  be  ai)plied  to  any  difference  of  even 
powers ;  thus, 


m 


ALUEBliAIC   OPERA  TIONS. 


and,  in  general,    d^^  —  ¥'^  =  (a"  +  b'^)  {a"  —  b"). 

If  tlic  exponent  is  a  multiple  of  4,  tlie  second  factor  can  be 
again  divided. 

EXAMPLES. 

a'-b*  =z  (rt2  +  b^)  (u^  -  b')  =  (rt'H^)  (a  +  b)  (a-b). 
a'-b'=  {a^  +  b'){a^-b')  =  {a'-}-b^)  {a'^  +  b-^  {a  +  b){a-b). 
When  b  is  equal  to  1  or  2,  the  forms  bectmio 

«2_1  -  (^a  +  l)(a-l). 

«3  -  4  =  (rt  +  2)  {a  -  2). 
rt2  4-  2a  4-  1  =  (rt  +  1)2. 
a2  ^  4a  +  4  =  (rt  +  2)1 
d'-2a-\-l  =  {a  -  1)2  =  (1  -  ay. 
cr2  _  4a.  4-  4  =  (a  -  2)^  =  (2  -  ay. 

By  putting  2b  for  Z>,  they  give 

a2  -  4/>»2  =  ^a  +  2i)  (a  -  2b). 
a2  +  4a^'  +  4^2  =  (a  +  2^)2. 

EXERCISES. 

Divide  the  following  expressions  into  as  many  factors  as 
possible  : 

A71S.  (a;2  +  4)  {x  +  2)  (re  —  2). 


4 

5 

7 

9 
II, 

13. 


1.  a;4_i6. 

2.  _?/♦  —  16a:4. 

3.  a;2  +  ()x  +  9. 
:c2  —  Ox  +  9. 
4a2.c2  —  dbY. 
92-2  -  12:r^  +  4^2. 
4a2a;2  +  iab.nj  +  l^y^- 

X*  —  2.1-y  +  ?/4. 

rt4  _  4^^2/^2  ^  4^4. 


^»5.  (a;  +  3)2. 


6.  ir.r^^.i-4— 1. 

8.  ff2<;2  -f  2a.r//  4-  yl 

10.  a*  +  ■\:dW  H-  4/A 

12.  a-4  —  4;r2//3  4- 4/. 


14.     «•' 


a262. 


a 


,2n 


'■^nH 


!a"  +  1. 


16.     x"^  —  4ax"  +  4a2. 


15- 

17.  1  — ?A 

18.  X^Z  4-  2.f3//3;j  +  /;?. 

Ahs.  z  {j^  4-  2.i-3/y3  -}-  /)  =  ^  (.r3  4-  iff. 


DIVISION. 


57 


20.     ci^  —  h^"-. 
22.     4r'^«  —  1J.<%2. 


19.     (fi—  ia%  +  4r^i2^ 

23.     4i-^//4  —  VZx^yi  +  U-f^. 

25.     .T»'"  —  'Zx'^'Uf  +  </-''».  26.     x^  —  ;>x'»»  +  1. 

1  .        „„  1 

27.     ic-  +  a;  + 


24.     x^  -^  x'if. 
-  2x^' 

28.     a;2»»  +  X'"  4- 


4  —     "       ■   "      -    4 

91.  By  combining  the  preceding  forms,  yet  other  forms 
may  be  found. 

For  example,  the  factors 

(a2  +  ah  +  ^2)  (,^3  _  f(j,  ^7,2)^  ( 1 ) 

arc  respectively  the  sum  and  dilference  of  the  quantities 

«'•  4-  b'    and    a/). 

Hence  the  product  (I)  is  e(|ual  to  the  diifcrence  of   the 
squares  of  these  (pumlities,  or  to 

Hence  the  latter  (|uantity  can  be  factored  as  follows: 
a»  +  «2^,3  +  Ji  =  (ai  +  aO  +  fj-')  {(e~  -  ah  +  h% 

EXERCISES. 

Factor 

I.  .r«  +  .'^•y  +  ?y*.                    2.  a^  +  8a7y2  -f  lOR 

3.  «* -f  !V?2/.' ^  81.^•^.               4.  rt^«  4- «'-'" /^2/i  ^  ^4n^ 

5.  rtU^  +  -i^rVA/'-'  +  l(V/'.i;2.       6.  (fi  +  8«'^2  _,_  lo^a^i. 

7.  .'?••'■'«  4-  a;3"  //-"  4-  .7;"//'". 

8.  w/2  _  ^8  4-  '■Zab  —  k     J«.s'.  (y»  —a-\-h)  {m  4-  ^  —  Z'). 
Here  tlio  last  three  terms  are  a  negative  square.     Compare  ij  83. 

9.  a^  —  4^2  4-  4bc  —  c\          1  o.  (i^  —  4r/Z''-  4-  4rtZ»6"  —  r/c-'. 

1)'5.  The  following  expression  occurs  in  investigating  the 
area  of  a  triangle  of  which  the  sides  are  given  : 


{((  -\-  h  +  r)  {a  +  b  —  r)  {a  —  b  +  c)  {n  —  b  —  c). 
By  §  80,  3,  the  ])roduct  of  the  first  pair  of  factors  is 
(a  4.  by  -  r-'  =  rt2  4-  2ab  +  I/i  —  c^ ; 

and  that  of  the  second  i)air, 

[a  -  by  -  r^  =  a--  2ab  +  b^  -  c^. 


(1) 


1 
i'  \ 

11) 

if 


-^ 


m 


ALGEBRAIC   OPERA  TIONS. 


!         ». 


By  the  same  priiicii)le,  the  product  of  these  products  is 

(^8  +  i^  —  c^f  -  ^aW, 

wliicli  we  readily  find  to  be 

«4  -\-h^  -\-c^  —  2«2Z>2  _  ^hh"^  —  2c^a\  (2) 

Ilenco  this  expression  (2)    can   be  divided  into  the  four 
factors  (I). 

Factors  of  Binomials. 
1)3.  Let  us  multiply 

OI'EUATION. 

X  —  a 


—  <y.Z;"-l  —  r/23;n-2  _  ^,3,.n-3 ^n   1^^  _  ^^n 


Prod.,  x^        0 


0 


0 


rt'* 


The  intermediate  terms  all  cancel  each  other  in  the  product, 
leaving  only  the  two  extreme  terms. 

The  i)ro.v;ict  of  the  niuUi})li('iind  by  x  —  a  is  therefore 
a:"  —  «".  llonce,  if  we  divide  x^  —  r/'*  by  x  —  a,  the  ([uotient 
will  be  the  above  expression.  Hence  the  binomial  x'^  —  «" 
may  be  factored  as  follows: 

a;«  —  a^  =  {x  —  a)  {x''-^-{-ax"-^  +  n'X''-^+ +rt«-2.f  +  ««-i). 

Therefore  we  luive, 

Tlieorem.  The  difforenco  of  any  powor  of  two  num- 
bers is  divisible  by  the  difference  of  the  numbers 
themselves. 

Illustuatiox.     The  difference  between   any  power  of  7  \ 

and    the   same   power  of  2   is   divisible   by  7  —  2  =  5.    For 

instance, 

r-  -  22  =r      45  =  5.9. 
73  _  23  _    3;j5  _  5.07. 

7^  -  2«  =  2:385  =  5.477. 
etc.  etc.  etc 


DIVISION. 


(2) 


a 


n-l 


For 


i 


94,  Let  us  multiply 

xn-\  _  axn-'i  ^  (i^^n-Z ^.  (_  r,)n-2^  ^  (_  ^^n-l 

by    a;  +  a  =  a;  —  (— «). 

Rem.  This  expression  is  exactly  like  the  preceding,  except 
that  —  rt  IS  substituted  for  a.  It  will  be  noticed  that  the 
cocfKcients  of  the  powers  of  x  in  the  multiplicand  are  the 
powers  of  —  a,  because  a 

(_rt)2  =  +a\ 

{-ay  =  +«*, 
etc.  etc. 

The  sigu  of  the  last  term  will  be  positive  or  negative, 
according  as  n  —  1  is  an  even  or  odd  number. 

OPERATION. 

X  -f-  a  =  X  —  {—  a) 

_1_  axn-i  _  a^jn-2  ^  (^s^n-a ...._(_ a)n-\x  _  ( _ «)« 
0 


Prod.,  x"         0 


0 


0 


The  multiplier  re  4-  «  is  the  same  as  x  —  {—  n)  (§  50). 
Ill  nuiltiplying  the  first  terms,  we  use  +  a,  and  in  the  last 
ones  —  (—  a)y  because  the  latter  shows  the  form  better. 

Hence,   reasoning  as  in   (1),  the  expression  «;«•—(—«)'* 
admits  of  being  factored  thus  : 
a;«  _  (_  rt)«  =  (re  +  «)  [.i-w-i  —  ax^-"^  +  a^x^-^  — 

+  (—  a^-^x  -f  (—  rt)"~M- 

If  n  is  an  even  number,  then  (—  «)»  =  «",  and. 
a;"  —  ( —  «)"  =  x^  —  rt". 

If  n  is  an  odd  number,  then  (—  «")  =  —  «",  and 

a"«  —  (—  (lY  =  a;»  +  a«. 
Therefore, 

Tfieorem  1.  Wlieii  n  is  odd,  the  biiiomiiil  x'^-^-W  is 
divisible  by  a;  +  a. 


f'^'MLffiniiiiinBaw 


60 


A LOEBHAIC   OPERA  TJONS. 


Tlieorem  2.  When  n  is  even,  the  biuomial  x^—a^  is 
divisible  \iy  x^-a. 

NoTK.  These  tlicorems  could  have  hecn  deduced  imme- 
diately from  that  of  §  *JIJ,  hy  clian^inj];  a  into  — «,  because 
X  —  a  would  then  have  been  changed  to  x  +  a,  and  x^  —  itP- 
to  x^  +  rt"  or  it"  —  oJ^y  according  as  n  was  odd  or  even. 

The  forms  of  the  factors  in  the  two  cases  are  : 

"When  n  is  odd, 
a;"  +  rt»  =  {x  +  u)  (;?'»-i  —  «.c«-«  +  rt2.c«-3  —  ....+  0"-!). 

When  w  is  even, 
a;"  —  a"  =  (a;  +  r^  (a^-i  —  rta:«-2  _|_  ^,2tn-3  _  . . . .  _<<n-i).      («) 

In  the  latter  case,  the  last  factor  can  still  be  divided,  be- 
cause a;"  —  «"  is  divisil)le  by  x  —  a  as  well  as  by  a;  +  a.     Wo 
find,  by  multiplication, 
{x  —  a)  (:r''-2  +  ^a.^n-i  ^  ^,i^^fi-6  +  . ...  +  r;n-2) 

Tlicrefore,  from  the  last  etjuation  {it)  wc  have : 
"When  n  is  even, 
x^  -  a"  =  (x  +  a)  {x—a)  {x''-^ -\- a-x"-*  +  a^x''-^  —  ....  +  «"-2). 

EXERCISES. 

Factor  the  following  expressions,  and  when  they  are  purely 
numerical,  prove  the  results. 

I.     52-22.  Ans.  (5  +  o)(5_2). 

[Proof.  52  —  22  =  25  —  4  =  21  ; 

(5 +  2)  (5 -2)  =  7-:J  =  21.] 


2. 

O-*  —  '4\ 

4- 

55  -  2'. 

6. 

73  +  23. 

8. 

1^  -  2*. 

10. 

a^  —  a\ 

12. 

a^  —  a^. 

14. 

x^  +  a\ 

16, 

8^3  _  27R 

18. 

;^-3  _|_  S//3. 

20. 

8«3  +  21/^3, 

3- 

5^  -  2\ 

5- 

5«  -  2«. 

7- 

73  -  23. 

9- 

.t2  —  a\ 

II. 

X*  —  a*. 

13* 

.t-3  +  ^3^ 

15- 

a^  —  8*3. 

17- 

10««  —  /A 

19. 

;*-»  -  1 «'.//. 

21. 

.<;«  -  Glk 

1 


fa 


DIVISION. 


61 


{a) 


LOfist  Coiiiinoii  ^Iiiltiple. 

05.  Def.  A  Common  Multiple  of  several  quanti- 
ties is  any  expression  of  which  all  the  quantities  are 
factors. 

Example.  The  expression  aiu^n^  is  a  common  nuiUiplo  of 
the  (lUiUitities  a,  m,  n,  am,  awn,  ai/i^,  m^n^,  etc.,  and  tinally  of 
tlie  expression  itself,  ain^/i'\  But  it  is  not  a  multiple  of  ^r*,  nor 
of  .r,  nor  of  any  other  symbol  which  does  not  enter  into  it  as  a 
factor. 

Drf.  The  Least  Common  Multiple  of  several 
quantities  is  the  common  mnlti])le  which  is  of  lowest 
degree.     It  is  written  for  shortness  L.  C.  ^I. 

BUI.R   FOR    FIXDING    THE    L.  C.  M.        FdCtuV    tJlC    SCVCrul 

quantifies  as  far  as  jmssible. 

If  tJie  qudntitics  have  no  cmmnoii  factor,  the  least 
comnioii  maUipJe  is  their  prmluct. 

If  several  of  the  quantities  have  a  coDiinon  faetor, 
the  inaJtij)le  required  is  the  jn'otluet  of  a/l  the  /'actors, 
earJh  of  theDi  I^eiiig  rtn'sed  to  the  hi^Jicsb  power  which  it 
has  ill  any  of  the  o'iveu  (piantitics. 

Ex.  I.     Let  the  given  quantities  be 

'Zab,         \Wr,         {jac. 

The  factors  arc  2,  IJ,  a,  b,  and  c.  The  highest  power  of  b  is 
I^,  while  a  and  c  only  enter  to  the  tlrst  power.     Hence, 

L.  C.  U.  =  Gat^c. 
Ex.  2.    rt2  _  I'i^  ffi  +  Oat,  +  b%  a"'  —  2ab  +  b^  a*  -  b\ 


Factoring,  wc  find  the  expressions  to  be, 
{a  +  b)  {a  -  Ir),     {a  +  bf,     {a  -  bf,     {a^  +  ^^M«  +  *)  (« 
By  the  rule,  the  L.  C.  M.  required  is 

{a-\-  bf{a-bY{a'  +  l^). 


b). 


i' 


ALGEBRAIC   Ol'FAtATIONS. 


EXERCISES. 

Find  the  L.  C.  M.  of 

I.  xy^  xz,  \jz.  2.    a%,  Wc,  (?d,  d^a. 

3.  a,  ab,  abc,  abctf.  4.     a^,  o.b^,  be*. 

5.  a;«  —  //-',  X  +11,  x  —  y. 

6.  .r*  —  4,  iT^  —  Ax  +  4,  .T?  -1-  4a;  -f  4. 

7.  IGrt'x^  —  4;;t',  '2^3*  f  w^  2rta:  —  m. 

8.  a:2  —  1,  x^  +  1,  a;2  —  2.r  f  1,  a;^  +  2.r  +  1. 

9.  4rt  (6  +  c),  b{a  —  c),  2ab. 

10.  2  (r«  —  Z')2,  2  {a  +  <:)-',  2  (a  —  b)  {a  +  b). 

11.  3ix  +  y),  3  :-)    SCr^  +  y^). 


12. 

13- 
14. 

15- 

16. 


a 


J,  rt2  _  ^v;      3  _ 


rt' 


Z**. 


a-'  —  a\  x^  +  fl3,  a:^  —  a%  x  -^  a. 
afi  —  G4a8,  X*  —  lGa\  x^  —  Aa\ 
a  +  b,  ««  +  2rtJ  4-  b"^,  a*  —  b*. 


Division  of  one  Polynomial  by  another. 

If  the  dividend  and  divisor  are  hoth  polynomials,  and  entire 
functions  of  the  same  symbol,  and  if  the  degree  of  the  numer- 
ator is  not  less  than  that  of  the  denominator,  a  division  may 
be  performed  and  a  remainder  obtained.  Tiie  method  of 
dividing  is  similar  to  long  division  in  Arithmetic. 

9(5.  Case  I.  When  fJirre  is  oiihj  one  algehraic  sym- 
bol ill  the  dividend  and  divisor. 

Let  us  perform  the  division, 

3ar»  —  4a;3  +  2a-2  +  3a:  —  1  -^  a^J  —  a;  +  1. 

We  first  find  the  quotient  of  the  highest  term  of  the  divi- 
sor .r^,  into  the  highest  term  of  the  dividend  ',\x\  multiidy  (he 
Avholo  divisor  by  the  fpiotient  'ix^,  and  subtract  the  product 
from  the  dividend.  We  repeat  the  })rocess  on  tlie  remainder, 
and  continue  doing  so  until  tlie  remainder  has  no  power  of  x 
80  high  as  the  highest  term  of  the  divisor.  The  work  is  most 
conveniently  arranged  as  follows: 


to 


DIVISION. 


63 


Sir'  X  Divisor, 

FirHt  Remainder, 
—X  X  Divisor, 

Sccoud  Itcinuindur, 
—8  X  DlvlBor, 

Tliird  and  last  Reinainder, 


Dividend. 

3.^-1  —  3.^3  4-  3.c2 


-     3^- 

:<;3  +  3a:  - 
a;8—    X 

-1 

1                '~~ 

'Ix^  4-  4x  - 
2a?«  +  22;  - 

-1 

Dlvlnor. 
32?*  —  a;  —  2       Quotient. 


22; +  1 


Tho  division  can  1)C  carriod  no  fartlicr  witlioiit  fractions, 
because  .r^  will  not  ^^o  into  x.  We  now  apply  the  .«anie  rule  as 
in  Arithmetic,  hy  adding  to  the  quotient  a  tract  ion  of  which 
the  numei'ator  is  the  remainder  and  the  denominator  the 
divisor.     The  result  is, 


3^«  _  4r3  +  22;2  +  3.r  —  1 


=  32;8  —  a;  —  2  + 


2.r  + 


{a) 


a;2  —  a;  +  1  a;^— 2;-,  1 

This  result  may  now  he  proved  hy  multiplying  th^  quotient 
by  the  divisor  and  adding  the  remainder. 

There  is  an  analogy  between  the  result  {a)  and  the  cor- 
resp(mding  one  of  Arithmetic.  An  algebraic  frac  -^w  like  (r/), 
in  which  the  degree  of  tho  numerator  is  greater  tiian  that  of 
the  denominator  may  be  called  an  improper  fraction.  As  in 
Arithmetic  an  improper  fraction  may  be  reduced  to  an  entire 
nuinJjer  plus  a  ])roper  fraction,  so  in  Algebra  an  im]m)j»er  frac- 
tion may  he  reduced  to  an  entire  function  of  a  symbol  plus  a 
proper  fi'action. 

EXERCISES. 

Execute  the  following  divisions,  and  reduce  the  quotients 
to  the  form  {a)  when  there  is  any  remainder. 

1.  Divido  a;^  —  22*  —  1  by  x  -\-  1. 

2.  Divide  x^  +  2a^'  —  2.c  —  1  by  a;  —  1. 

3.  Divide  j"^  —  ^x^  4-  2a:  —  1  by  a?  —  x. 
2x^  4-  x^  —  a;  —  5 


4.   Reduce 
5 


0,-t 

/».' 


a^  —  X  -^  i 
Divide  24^3  —  SSa^  —  :]2a  4-  50  by  2a  —  3. 

Ana.  Qiiot.  =  i:la^  —  a  —  \'  ;    Keni.  =  — 


): 


04 


ALU  Eli  UA IV   OP  Eli.  1 TIONS. 


*• 


6.  Divide  X*  —  1  by  .c  —  1. 

Wlu'ii  terms  are  wanting  in  the  tlividcnd,  tli(>y  may  bo  conHidcrcd  hh 
z«'m.  In  this  la.st  cxcrt'if^c,  the  terms  in  ./■',  .r',  and  ./■  are  wanting;.  l?iit 
the  U'ginner  may  write  the  dividend  and  i»erl'orin  th(!  operation  tiiuti : 

jc^  -t  O./"'  +  0/''  +  Ox-  -  1    \  x-\ 


a-*- 

ar^ 

x"  +  Ox'^ 

a;-  +  0.C 

J--'  —     X 

x-X 

x-\ 

0     6 

Tlio  oi)(>rntien  is  thus  nssimihited  to  tliut  in  which  the  expression  is 
cnnipietf  ;  hut  tlie  aMual  writinj;  oC  tlie  zero  terms  in  tills  way  is  iin- 
net'essary,  and  sliould  bo  dispensed  with  as  soon  at)  the  btudent  is  uIjIo 
tu  do  it. 

7.  Divide  a^  —  'la  +  1  by  a  —  1. 

8.  Divide  x^  +  1  by  x+  I. 

9.  Divide  8f<^  +  Vi'y  by  '4a  4  5. 
Divide  a^  +  1  by  a  -{-  \. 
Divide  a^  +  'ia-  +  I>  by  a"  +  2rf  +  3. 
Divide  ««  —  1  by  a^  +  'Za'^  +  2rt  +  1. 


10. 
II. 
12. 

13- 


Divide  :r«  —  !:>./'  +  SG.r*  —  .'J'-i  by  tf=«  —  2. 


14.   Divide  (j^  —  2j!:  4-  1)  (•'''  —  l''-''  —  D>)  by  a:-  —  10. 

For  some  pur|)oses,  we  may  ecjually  well  perform  the  operation  hy 
hcfrinniii;;:  with  the  term  containing  the  lowest  ])ower  of  the  quantity, 
or  not  containing  it  at  all.     Tak<',  for  instance,  Kxample  9  : 

12")  +  8^t3     \Jy±'>' 
12.">  +  oOfl! 

-ma 

-  'Ma  -  20rt' 


In 


25  -  \0a  +  ia' 


20a'  +  8a^ 

15.  Divide  1  +  3.T  +  3.i-'  +  x^  by  1  +  x. 

16.  Divide  1  —  4.C  4-  4.c«  —  x^  by  1  —  r. 

17.  Divide  15  +  'la  —  oa^  +  a^  +  2^/^  —  a-'  Ity  5  +  4rt 

18.  Divide  1  —  //'  by  1  +  2i/  +  If  -f  f. 

19.  Divide  (il  — (Mz-fKU-— .S.f=*  +  4.r*— .<•'•  by  _4  +  2.c-|-.c2. 

20.  Divide  G-4  —  IG./^  -f-  :<;6  by  4  —  ix  +  a?^. 


rt''. 


DIVISION. 


69 


a\ 


07.  Cask  TI.  ^Wtrn,  tJirrv  arc  several  algebraic  syni- 
bols  hi  the  divisor  diul  dividend. 

Let  us  suppose  the  dividend  and  divisor  nrranfred  acrordin*:^ 
to  powers  of  some  one  at"  tjjc  symbols,  uliicli  we  may  sui)i)oso 
to  be  X,  as  in  >J  70. 

Tjot  us  eall  .1  the  coetlieient  of  the  highest  power  of  x  in 
the  dividend,  and  //  the  term  independent  of  x,  so  that  the 
dividend  is  of  the  Ibrm 

yla;"  +  (terms  with  lower  powers  of  j)  +  //. 

Lot  us  call  a  the  coeflieient  of  the  highest  power  of  x  in 
the  divisor,  and  h  the  term  of  the  divisor  independent  of  x,  so 
that  the  divisor  is  of  the  form 

ax"^  -\-  (terms  w  ith  lower  powers  of  x)  +  h. 

Then  we  have  the  following 

Tlivorem.  In  order  that  the  dividend  may  be  exact- 
ly divisible  by  the  divisor,  it  is  necessary  : 

1.  That  the  term  containing  the  higliest  power  of  x 
in  the  dividend  shall  be  exactly  divisible  by  the  cor- 
responding term  of  the  divisor, 

2.  That  the  term  inde])endent  of  x  in  the  dividend 
shall  be  exactly  divisible  by  the  corresponding  term  of 
the  divisor. 

Jienson.  The  reason  of  this  theorem  is  that  if  we  suppose 
the  quotient  also  arranged  according  to  the  powers  of  .r.  then, 

1.  The  highest  term  of  the  dividend,  .1./",  will  be  given  l»y 
multiplying  the  highest  term  of  the  divisor,  nuf^,  by  the  high- 
est term  of  the  quotient.     Hence  we  must  have, 

Highest  term  of  quotient  =  -      • 

2.  The  lowest  term  of  the  dividend  will  be  given  by  multi- 
plying the  lowest  term  of  the  dividend  by  the  lowest  term  of 
the  <iuotient.     Hence,  we  must  have. 


Lowest  term  of  quotient  = 


ff 


llE^i.  1.     Since  we  may  arrange  the  dividend  and  divisor 
according  to  the  jwwers  of  any  one  of  the  ^symbols,  the  aljove 


b 


60 


ALQICDRMC  OPEIiATIONS. 


tlu'ore'm  must  Ijc  true  whatever  symbol  we  tuko  in  tlio  place 
of  ;c. 

UvM.  2.  It  (loos  not  follow  that  when  the  conditions  of 
the  theorem  are  fulfilled,  thf  division  can  always  be  performed. 
This  (|uestion  can  be  decided  only  by  trial. 

Wc  now  reach  the  folhjwing  rule : 

I.  Arrange  both  diiudcnd  ami  du'isov  acmrdin'J  to 
the  (tsccndinjj  or  dcsccmlinjj  puivcrs  of  some  com  in  on 
si/nihol. 

II.  Fowl  the  frst  term  of  the  quotieut  hi/  (ViviiViii;^ 
the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor. 

III.  MiiUiphj  the  jrhole  divisor  bij  the  term  thus 
found,  and  sulttroet  the  prod  net  from  the  dividend. 

IV.  Treat  the  remainder  as  n  new  dividend  in  the 
same  way,  and  repeat  the  process  until  a  remainder  is 
found  ivhieh  is  not  divisible  by  the  qiwtient. 

Ex.  I.    Divide  ofi  +  Zax^  +  ^a^x  +  a?  h-^  x -\- a, 

OPERATION. 

Cf^  -|_  ^ax^  +  3rt2x  +  rt3  \x-irn 


0^  -f-    aji^ 


2ax^  +  3^22; 
2ax^  +  2a^x 


z^  +  2ax  +  «« 


ci^t  4-  a^ 
o^x  4-  rt^ 


0        0 
Ex.  2.    Divide  7?  —  a:i^  -\-  a  {h -\- c) x  —  ahc—hx^—cx^^lcx 
by  a:  —  a. 

Arranging  according  to  §  76,  we  have  the  dividend  as  follows; 

.T^  —  («  4-  i  +  c)  x^  +  {ah  ■\-l)C -\- ca)  x  —  aha  \x  —  a 
a^  — 


ax^ 


—        {h-\-c)  x"^  +  {(lb  +  hc  +  ca)  x 
— {b  -\-c)x^  -\-  {ab  4-  ac)  x 


x^  —  {b  +  c)  X  -\-  be 


hex  —  abc 
hex  —  abc 


I. 

2. 

3- 
4- 

5- 


FRACTIONS. 


07 


1.  Divide 

2.  Divide 

3.  Divide 

4.  Divide 

5.  Divide 

6.  Divide 

7.  Divide 

8.  Divide 

9.  Divide 

10.  Divide 

11.  Divide 

»2.  Divide 

13.  Divide 


EXERCISES. 

the  dividend  of  Ex.  2  above  hy  x  —  b, 

tiie  dividend  of  Kx.  2  ubove  l)y  ;/•  —  c, 

<i^  -f  /y  —  r-3  4-  '.hibc  l)y  rt  +  &  —  r. 

a^  +  i''  +  Dab  —  1  by  a  +  />»  —  1. 

aV/^  +  'inbx^  —  (r/«  f  ^)  r^  by  ab  +  {n  —  h)  x. 

[,{i  _  hcf  -f  H/»V  by  rt'^  +  be. 

{a  +  b  +  r)  {(lb  -f-  be  -f  m)  —  abc  by  rt  +  J. 

(^f  -f-  i  —  c)  {b  +  c  —  a)  (e  4-  «  —  ''O 

by  rt»  —  ^rJ  —  c2  -f  2*c. 
r/''  +  Zr»  +  c3  —  3rtJc  by  a  4-  ^  4-  c. 
.r»  4-  4^«<  by  a;''-*  —  2ax  4-  ^a'. 
rt'J  (b  ^x)  —  bii{x  —  a)  +  {a  —  b)x^  +  abx 

by  a;  4-  a  4-  5. 
T^  —  ax^  —  />2r  4-  aP  ])y  (a-  —  a)  (r  4-  /;). 
I;ia^.f9  —  lAn'^x^  4- 1^«''.<'-^  —  a'  by  ^a'^x^  —  a^. 


m 


-♦-♦-♦- 


^J/c 


CHAPTER     IV. 
OF    ALGEBRAIC     FRACTIONS. 

OS.  Drf.  An  Algebraic  Fraction  is  tlie  expression 
of  an  indicated  qnotient  when  the  divisor  will  not  ex- 
actly divide  the  dividend. 

Example.     The  quotient  oi  p  -—  q  is  the  fraction  -• 

Drf.  The  numerator  and  denominator  of  a  frac- 
tion are  called  its  two  Terms. 

Trnusforniation  of  Single  Fractions. 

99.  Rediiotlon  to  Loicesf  Terms.  If  the  two  tenns 
of  a  fraction  are  multi])lied  or  divided  by  the  same 
quantity,  the  value  of  the  fraction  will  not  be  altered. 


:|i 


/, 


G8 


ALGEliRAW  OPKUA  TI0N8. 


(IX 

Example.     Coneidcr  the  fraction       •    11'  \vc  diviik'  both 

X 

teniis  ^)y  a,  the  fraction  will  become  -• 

y 

ax  _x 

«y  ~  y 

Corolhirif.  If  the  miiiuTator  and  (L^noiiiinator  con- 
tain cunmion  factors,  tlicy  may  be  cancelled. 

Def.  Wlien  all  tlie  factors  common  to  the  two 
lerms  of  a  fraction  are  cancelled,  the  fraction  is  said  to 
Ibe  reduced  to  it.s  Ixj-west  Terms. 

To  reduce  a  fvnrtimi  to  its  hticcfit  tennis,  factor  both 
terms,  irhcii  ncccsstiry,  and  cancel  all  Ike  common 
factors. 


Ex.  I. 


acmf 


en 


Ex.  2. 


The  factor  inf  comraou  to  both  terms  is  cancelled. 
(m  _  n^ 

The  factor  d^l?  common  to  loth  terms  is  cancelled. 

Ex.  \.     Itt'dnce  —.-• 
a'x 

Here  a^x  is  a  divisor  of  both  terms  of  the  fraction.     Di- 


a\c 


vidiiiff  l)y  it,  the  result  is  -,•    Ilencc     ~-  = 


a' 


Ex.  4. 
Ex.  5. 


(i^  +  Ub  -Ju  I? 


a*x 


1^ 

rt2 


a  -f  1} 


«'  —  i/i  {a  +  b){a  —  b)       a  —  b 

mu  —  nu  (m  —  n)u  u 

mx  —  nx  ~  (m  —  n)  x  ~~  x 


^ 


EXERCISES. 

Reduce  the  followinci  fractions  to  their  lowest  terms  : 


I. 


1 0/irfi^ 
l-Jyry-*" 


2. 


am 


1  'iff.ri/ 
1  ^ui'ir^t/'^ 


M 


FRACTIONS. 


i  both 


'  con- 

I  two 
id  to 

•  hofJi 
Linuib 


I)i- 


5- 
7- 
9- 
II.     —. 

13- 
IS- 
17- 


7-)  (r,  _^7-)  (/^— r)_ 

aji  —  hy 
ax  —  ba- 

d^-1fi 

u'i-iab  +  1?' 

a^  -\-  y^ 
a{x~-^  y)' 
a^-^b* 

a^  —  y^ 

axm  —  axn 
by  HI  —  by  II 


6. 
8. 

10. 
12. 
14. 
16. 

18. 


69 

__20(f( -]- x)(m —  }f) 
24:  {d^—2ax  +  x^){m  —  u) 

ay  — by  ' 
d^  +  4ax  +  4a^ 
a^—J:^ 

_ — . « 

ay  4-  ^^/y 
rt'^  +  ab  +  /»2^ 
«4  ^^;.^2  _^-^4* 


aw 


X 


an 


7;/.r  —  7?  a; 


(«  H-  b)  {m  —  w) 


100.  Rnh  of  Sifpi!^  in  Fraclion.'^.  Since  a  fraction  is  an 
indicutod  ([uotient,  the  rule  ol'  si<,'ns  corresponds  to  that  for 
division.  The  followini^  theorems  ibllow  from  the  laws  of 
ninltiiilication  and  division: 

1.  If  the  tcniiH  'AY(\  of  tlie  same  sign,  tlio  fraction  is 
positive ;  if  of  ()i)i)osito  signs,  it  is  negative. 

2.  Cyiumging  tin'  sign  of  either  term  changes  tlio 
sign  of  the  fraction. 

3.  Changing  the  signs  of  both  terms  leaves  tlie  frac- 
tion v»itli  its  ori^',inal  sign. 

4.  Tlu^  sign  of  the  traction  may  bo  changed  by 
changing  tlie  sign  written  l)efore  it. 

5.  To  these  may  be?  added  thegenei'al  princijde  that 
an  even  nnmber  of  changes  of  sign  restores  the  fraction 
to  its  oiiginal  sign. 


Ex.  I. 
Ex.  2. 
Ex.  3 


n        — 

a 

—  a              a 

b~^ 

~b 

-  ~    b     "  ~  -b 

n 

—  n        —a          a 

~b  ~ 

'  -b  ~  T  ~  -  b' 

a  —  b 

b  —  a              a  —  h 

b 

—  a 



^^~^ 

-^2.    —                  ■   r::^ 

__  — 



m  —  n 

;/  —  III             n  —  III 

m 

—  u 

l> 


70 


ALOEBRATC   OPERATIONS. 


EXERCISES. 

Express  tlie  following  fractious  in  four  different  ways  witjj 
resi)L'ct  to  si;:^us : 


I. 


5-     - 


X 

a 
m 

V 

— 

!7 

m 

—  n 

2. 

4- 
6. 


X  —  1/ 


a 

— 

b 
a 

- 

a 

— 

b 

+ 

c 

a 

4- 

m 

X 

p  +  q  —  r  a  —  in  -\-  x 

Write  the  followinir  fnictions  so  that  the  symbols  a;  and  y 
shall  be  positive  in  both  terms  : 
X  —  b 
c  —  y 
a  -{-  X  —  /; 
a  —  X  -\-  b 
X  —  a  -\-b 


7- 

9- 


+ 
+ 


8. 


lO. 


II.      — 


n  - 

-X 

• 

-y 

a  — 

X 

b- 

X 

a  +  b- 

—  X 

a  —  b  +  y 


b  —  X 

101.  Whon  the  numerator  is  a  product,  any  one  or 

more  of  its  factors  can  be  removed  from  the  numerator 

and  made  a  multiplier. 

abmx          ,    mx           ^         on  ,1 
=  a6  ~      -  =  abm =:  abmx — 


Ex. 


P  +  ^I 


=  abmx 
p  +  q  V-Vq 


EXERCISES. 

Express  the  following  fractions  in  as  many  forms  as  possi- 


I. 


bio  with  respect  to  factors  : 

pi'X  ab 

'         ,  2.         - 

1)\  )h  C 

rf2  y-i  ffi 

ir^b'        5.  - 


abc 
3-     a  +  b 


b* 


X 


X* 

6.     -- 


IGa* 


X  -\-  :ia 

103.  Ilediictlon  to  Given  Benom'niator.  A  quan- 
tity may  be  expressed  as  a  fraction  with  any  required 
denominator,  i>,  by  sui)po8ing  it  to  have  tlie  denomi- 
nator 1,  and  then  niulti])lying  both  terms  by  I). 

For,  if  wo  call  a  the  (quantity,  we  have    a  =  .-  —  -.-\ ' 


FRACTIONS. 


71 


Ex.  If  wc  wisli  to  express  tlic  quantity  ah  as  a  fraction 
having  xy  for  its  dencjminator,  we  write 

(ilKiy^ 
xy  ' 

If  the  quantity  is  fractional,  both  torms  of  tlie 
fraction  must  be  multiplied  by  that  factor  which  will 
produce  the  requirc^d  denominator. 

Ex.  To  express  .-  with  the  denominator  n¥,  we  multii)ly 
both  members  by  nb'^  -h  i  =  nl^.     Thus, 

a       aul)^ 

b  7lb^ 

This  process  is  the  reverse  of  reduction  to  lowest  terms. 

EXERCISES. 

Express  the  quantity 

I.     a  with  the  denominator  b. 


2. 

ax 

u 

u 

a 

ax. 

3- 

ab 

» 

U 

« 

ab'K 

4- 

m 
n 

u 

u 

(( 

n  (x  —  y). 

5- 

-1 

t( 

it 

M 

m* 

6. 

w  {n  —  p) 

a  +  b 

a 

a 

(( 

a?  -  b\ 

7- 

x  +  y 
X  —  y 

« 

11 

it 

x^  -  y\ 

8. 

x^  +'l 
X  +1 

(( 

« 

it 

3-2  +  2.T  +   1. 

0. 

«  +  l 

(( 

(( 

<( 

a^  —  1. 

a-i 

Nojifativc  ExpoiH'iits. 

103,    13y  the  principle  of  i<  S5,  we  have 

If  we  have  k  >  n,  the  exponent  of  the  second  nieml)cr  of 
the  equation  will  be  negative,  and  the  first  member,  by  can- 


h\ 


ALaEBRAia   OPERATIONS. 

eolliiif^  n  factors  from  each  torni  ol'  the  fraction,  will  Lcomo 

1 


ikn 


i^-" 


—  «»  *. 


i>y  ^Hitting  for  sliortneHS  k  —  n=.  s,  the  equation  will  bo 


a* 


He  lire, 

A  n('!j((tin'  cxpnvcnt  iiulirdtes  the  rrri/n'orrtJ  of  the 
co?'rcfif)on(lini,'  (/iKiiUiti/  itith  a,  positiue  exponent. 


a 


If  ill  the  formula  «""*  =r    v    we   suppose  k  =  n,  it  will 


a 


<i' 


Leconie  a"  =:   ^  ,  or  r/"  ■—  1.     Jlence,  because  a  may  be  any 
quantity  whatever, 

Any  qaaiititij  with  the  exponent  0  in  equal  to  nnity. 

Tills  result  may  1)0  nuule  more  clear  by  suc- 
cessivf  divisions  of  n  powiT  of  (t  by  a.  Kvory 
tliii"  wo  crti'ct  this  division,  we  diminish  the  cx- 
pontMit  by  1,  luul  \vi'  may  suppose;  this  diminution 
to  continue  aljrebrainilly  to  nepitive  values  of 
the  exponent.  On  the  left  hand  Hi(U'  of  th« 
rcpiatioMs  in  tlie  mar^dn,  the  division  is  efVected 
Hymlxiiically  by  diminishing  tlu*  ex|)onents;  on 
the  right  tlie  result  ia  written  out  in  the  usual 
waj. 

EXERCISES. 

In  the  following  exercisi's,  write  t),,-  juotients  which  am 
fractional  both  a.s  fractions  rciluced  tc;  fh*  ir  lowest  terms,  wwX 
OH  entire  <juantities  with  negative  exponents,  on  the  j)rinciple. 


(l^    = 

a<«i 

«2       =: 

aa 

r/l       = 

a 

ao     = 

1 

a-1  = 

1 
a 

«-a  = 

1 

an 

etc. 

etc. 

a 


(I 


=   (^^'\         yi    ==    «^^~^       etc. 


I.  ^  by  ;«r. 

t.  0  by  ufl. 

3.  —  2lr'  by  /A 

4.  4,,r  l,v    -?,>(/«*. 


-/l«.s'.       or  .t~'. 


^,,.9.  _  ^ ,.   or  -  2rra/y2. 


I' 


tern 


FiiAnrioNs. 


73 


IS,   Mll'l 

luiple, 


x-\ 


9 

lo 

II 

12 

13 

M 

IS 
i6 

17 

i8 

19 


6.     V]a^/P.ri/  hy   Uthx. 


2Aapif.i;/  l»y  Ibuuc. 


—  SftV)  l)y   Irr/r. 
14r<VA-3  by  -  7«2^'f».         { 

—  'MUiy.c'-t/  l)y  —  24(;^r/y. 
4Srt^  (a;  -  7/)-'  by  uG  (:f  -  //). 

22  {a  —  Z»)  (yy<  —  ;/)  by  15  (a  +  />)  (m  +  -V 
25  (a2  —  ^/^)  (;//2  _  ji-i)  by  15  (r«  -  />)  (/yi  +  ^O- 

{X-*  -  1)  (rt2  -  4^2)    Ijy    (;c2  _  1)  ^(i  +   JJ^). 

a:6  —  1  by  .f3  +  1.      . 

«-^.37^^5  by  rt'''(!/».r''y/3. 

m^ti^yh  by  mn^y^A 

m  {m  4- 1 )  (//i  +  2)  (///+;>)  l>y  w  (m— 1)  (^/?— 2)  (w  — 3). 

a"'  by  «".  20.     «i"'6n  by  ql'^c'^K 

Diss<»<*tu)ii  <)f  Fractions. 


104.  If  the  mimerator  is  a  ])c)lyn()mial,  racli  of  its 
t<M-nis  may  bo  divided  Heparatcly  l)y  tlie  denominator, 
and  tlie  several  fractions  connected  by  the  signs  +  or  — . 

The  ])riiu'i])le  is  that  on  wliich  the  division  of  polynomials 
is  founded  (5^  ST).     Tlie  general  form  is 

A  +  B  -^-C -\-  etc. 


m 


A     B     a      ^ 

-  +  -+-  +  etc. 

Vl         111         III 


(I) 


The  separate  fractions  may  then  be  reduced  to  tlieir  lowest 
terms. 

Example.     Dissect  the  fraction 

^^Iqihh'  —  ISaun/  4-  15/y»z  — 12&Vt^_ 
Wadx 


The  general  form  (1)  gives  for  the  sei)arate  fractio«», 

d-idVAr.       mi  mi/    ,    Wbiiz        ni?}iht 
lijabx 


+ 


ICutbx     '    lintbx         \^\abx 
Reducing  cacli  fraction  to  its  lowest  terms,  the  sum  becomes 

Unhv 

—  ■■  —  —  • 


2iib 


Umj/       ]5«2 
bbx       idax 


T4  ALOEPnATC   OPERATIONS. 

EXERCISES. 

Separalc  into  .'iiins  <»f  t'nictions, 

abc  -f-  bed  +  nla  -f  dab 

abed 

—  XIJZU  -f-  J-\l/ZN'  -f  Xl/Z'U  —  jfijI^Z^U^ 


a^-h^ 


xhfzhi^ 


(fix  —  b'j/ 


ab  ax 

{m  —  v)  {n  +  q)  —  (m  -{-n){p  —  g) ^ 

(^ui  —  n)  {p  —  q) 
{x  -  a)  {y-b)  +  {x  -  ij)  {a  -  b)  +  (x 


b)  0/  -  a) 


x^  —  y"^ 
{a  +  b)  (>n  —  ;/)  —  (a  —  b)  {in  +  n) 
a3  -> 


Agrsrrosratioii  of  Fractions. 


^r>?^ 


105.  Wlien  several  fractions  liave  equal  denomina- 
tors, tlieir  sum  maybe  ex])ressed  as  a  single  fraction 
by  aggregating:  their  numerators  and  writing  the  com- 
mon denominator  under  them. 

A       II      a       A  —  B-^a 

Lx.  I.       1-       = • 

ni       111       III  m 

T-,  a  —  b      b  —  c       c  —  a 

Ex.  2.        "  -1-      -  -f        — 

x  —  y     y  —  J^     -c  —  y 

a  —  b      c  —  b       c  —  a        2e  —  2b       2(c  —  b) 
~  x  —  y      x  —  y      x  —  y~'x  —  y~     x  -  y 
Kem.    This  jtroct'.SH  is  the  reversn  of  that  of  dissecting  a  t'luction. 


EXSRCISES. 

Aggregate 

a        lib       abc                           a 

-{n-bY' 

a^c      abc   '   abc                *"     (a  —  bf 

^      x  —  ay^-ba  +  bx  —  y 
^'       a'x     ^     n\c    "*"     a'^x    '^     a^x 

ah              c              d 
a  —  b      b  —  a      a  —  b      b  —  a 

a  —  b        a  —  c        c  ~b    ,    c  -\-  a 

S«                                                             ■                 T                    ■ 

m  a 


m  ■—  n      ill  —  n      n  —  m      n  —  m 


FRACTIONS. 


75 


(i) 


100.  Wlion  all  tlio  frnetious  havo  not  tlio  pamo  (lonomina- 
tor,  llu-y  iiiiist  l)c  rt'diiced  tu  a  cummoii  (k'nomiiiator  by  tlio 
process  of  i^  lO::^. 

Any  conmion  iniiltiplc  of  the  dinoininators  may  bo  takon 
as  the  comniou  deuouiiuutor,  but  the  leuiit  cuuimon  multiple  is 
the  simplest. 

To     UEDUCK    TO     A     COMMON     DeXOMINATOU.      CJlOOSC   (I 

coDinion.  niuUiple  of  the  dcnouiiiiatorfi. 

MiUliphj  hotli  terms  of  each  fvdction  hij  the  witJti- 
plier  iicvesfidvij  to  change  its  dcnoinimUor  to  tJic  chosen 
viuJti//lc. 

Note  1.  The  rofniired  multii»liers  will  be  the  finoiionls  of 
the  chosen  multiple  by  the  denominator  of  each  separate 
fruetion. 

Note  H.  AV'hcn  the  denominators  have  no  common  fac- 
tors, the  niultii)lier  for  each  fraction  will  be  the  ju'oduct  of  the 
denominators  of  all  the  other  fractions. 

Note  ',].  An  entire  (|uantity  must  bo  regarded  as  having 
the  denominator  1.     (j<  10"^.) 


---  • 


EXAMPLES. 

1.     Af?Ln*c']rate  the  sum 

J  _1       1  __1     ,    _!_ 
a       lib      ahc      abed 
in  a  single  fraction. 

The  least  common  multii)le  of  the  denominators  is  nhcil. 
The  separate  multipliers  necessary  to  reduce  to  this  com- 
mon denominator  arc 

abed,     bed,     cd,    d,    1. 

The  fractions  reduced  to  the  common  denominator  «Z»Cf/  ure 

abed      —  Ited       -\-  ed      —  d       -\-  \ 
ubed^      abed  ^     abed*    abed*    abed 

abed  —  bed  -f  cd  —  d  -\-  I 


The  sum  is 


abed 


By  disscctin.tj;  this  fraction  as  in  §  104,  it  may  be  reduced 
to  its  oriixinal  foiin. 


76 


A  L  a  imUA  IC  OPhJlU  TIONS. 


2.  Keiluce  tlie  siuii 

1       a      b      r 
abed 
to  a  single  fraction. 

'The  nuiltiplifn  are.  by  Note  2,  bed,  anl,  abil,  abc. 

XJ:5iiig  tiicse  multipliers,  the  I'niclionis  become 
bnl,        —  (Ord      all'd       —  (dx^ 
abed*       abed  '     abeil^       abed  * 

from  Avbicb  \hv  nM|uiro(l  snm  is  readily  formed. 
.'^.   lieduce  the  snm 

Tbc  least  common  miiltii)le  of  tlic  denominators  is  x^ 
The  mnltipiiers  are,  by  Note  1, 

^•2—1,    x-{-l,     .r  — 1,     1. 

The  snm  of  the  fractions  is  fonnd  to  be 

;,2  _  1   _,..,.  4.  1   4_  j.i  _  ,,.  +  3.2  ;).,2 


-1. 


x^ 


x^ 


EXERCISES. 

Kcdnce  to  a  single  fraction  the  sums, 

I.      1  4-  ;•  2.      1 ---. 

.<■  —  1  X  -j-  1 

1  —  X    r+  x'         ^'  '    -  + 


3 

5.  ^ 
7. 

9- 


10. 


ax  x^ 

a  -\-  X      a  +  X 
a  X 


X  {a  —  x)       n  {a  —  x) 
1  2?/  1 


6. 


8. 


i  —X       1  -i-  X 
a  b 


+  -i2 


a;  -I-  y/        X-  —  y        X  —  IJ 
_1  I  _  1 

r/  —  b      b  —  c      c  —  a 


a  a 

II.     — ,~  + 


x-\-  ij      x  —  y 


12. 


a  —  b       a  -\-  b 


'Zx  —  5 


+  0 


5 


4x-2— 1    '   "Zx  —  l 


3 

—  • 

a; 


a  +  b      n  —  b 
a  —  b      a  \-  b 


FRACTIONS. 


77 


3 

X 


13- 
14. 

15- 

16. 

18. 
19. 
20. 
21. 
22, 

23- 
24. 

25' 

26. 
27. 
28. 
29. 

30- 


i  rt 


di—i;i  ~  a  —  b  '   a  ■{-  b 

1 1__   _  1 

^  {x  -  1)        'Z{x  +  I)       ;^ 

a  I .  h 

a  —  b 


(-.-.)• 


m  -\-  n      X  —  // 

1)1  — 

1  - 


m  —  n      X  +  y 
a 


17. 


V       *"  +  y 


m^ 


m  {)u  —  I/) 


3-2 


a  —  X      d^  —  x^ 


a  —  b       b  —  c       c  —  a       (a  —  h)  {h  —  c)  {c  —  a) 
b 


a       I     b       ,       f^    \ 
0  ~\a  -b  ^b~-^il' 


m  —  {x  —  (i) 

^+  y  ~ 

ab      be      iic 


m  —  {x  ■\-  (i) 
X  —  y  ~' 


a 


-^    + 


b 


+ 


{a  —  b)  [a  —  c)    '    {b  —  a)  {b  —  c)    '   {c  —  a)  {c  —  b) 
X  +  I       X  —  I 


X  —  I       X  -\-  1 


+  ix. 


ab  r/2  a  (^/^  4.  //i) 


a  -\-  b      a 
a 


1 
1 


(j  "^"  ^~di\-  (ji 


X 


X  -\-  a      X  —  a 

.'C2  —  2:r?y  -j-  yi 


:<;2  4-  y 


[3 


1    _^l.'tJf 

'Zay 
1 


,2  _  .,.2 


(,f   +   A)2     ■     (,<  _  /,)2  +  ^f2  _  ^2 

^2  _  '^)r?/;  +  /;2 


1     r 


M 


III! 


■iaii 


78 


ALOEIUIAW   OPKltA  TIONS. 


Factoring?  FrJU'tioiiH. 

107.  If  Rovcml  tcima  of  tlio  immorator  contain  a 
romiMon  fjictor,  the  coclIiciciitH  of  this  fjictor  iiuiy  bo 
added,  and  their  a^jj^iv^ate  niuitiplicd  by  the  factor  for 
a  nuw  form  of  the  numerator. 

EXAMPLES. 

ax  —  b.r  +  ex  -\-  dx        {<i  —  h  -\-  r  +  d)  x 


I. 


m 


m 


X 


2. 


=  {a-b  +  c-\-d)~.    (§101.) 
ohx  -f  hex  -f  nnjf  —  ahf/ (ah  ■{■  hr)  x       {ne  —  nh)  y 


ahn 


obn 


abn 


=  {a  +  c)~  +  ic-b)f  ' 


EXERCISES, 


3 

4 

S 
6 

8 
9 

lO 


Reduce 

abi/  —  bry  —  acy 

abq  +  brq  -\-  abr  4-  brr 


mnn  +  mpu  -f-  pun 
I  an 


abc 

ax  —  /;//  —  3J.r  —  iay 


'4  ma 


4nix  -\~  2y  —  3ax  —  C)rx  +  ay 

a^  4-  2^2*  -f  al^  a^x  —  iabr  —  (3?/  —  4r)  n 

.  y, ■ . 

xy  I)  +  Y 

ygy  —  [4.g  +  X  (2b  —  Ac)  4-  3ry.r] 

rt.r'  —  \r,x  —  3  [/??.r  -f  w  {a  —  x)  —  am'\ 
_._  _  ^^^  ^^  _^^^ 

4a  ^x  —  2f;\/;r  +  2b^/x  —  2  (nniVx  —  4\/y). 


II( 


I. 


FRACTIONS. 


It 


3fii1tii>1i<'ati<>ii  and  Division  of  FnicnouM. 

lOS.  Fundamental  Tluorems  lu  the  Mull iidlattUni 
and  Dlcision  of  Fr actions : 

Theorem  I.  A  fractifui  may  \w  imilti])li((l  bv  any 
quantity  by  ritljcrinultipl^in^Mts  miiueratur  or  dividing 
its  duiioniiimtor  by  that  (juaiitity. 

Cor.  1.  A  fniclioii  may  bo  multiplied  by  its  (leiiominator 
by  simply  cuncolling  it. 

L'ur.'l.  If  the  ilenominator  of  the  fraetioTi  is  a  fnetor  in 
the  multiplier,  oaiurl  the  denominator  to  multiitly  by  this 
factor,  and  then  multiply  the  numerator  l>y  the  other  factors. 

Ex.        --  " -7^  X  ^'-  (./•«  -  IP)  =  am  {x  +  i), 
a  {x  —  a)  ^ 

because  the  mult  i^jlier  a'  {x"^  —  IP)  =  a{x  —  h)  a  (x  +  b). 

Theorem,  II.  A  fraction  may  be  dividiul  by  rithcr 
dividing  its  nnnn'rator  or  multiidying  its  denominator. 

Th(i>rem  III.  To  multii>ly  by  a  fraction,  the  midti- 
plicand  must  bo  multi))li<'d  by  the  numerator  of  the 
fraction,  and  this  j)roduct  must  bu  divided  by  its  de- 
nominator. 

Lot  us  MUlltiplv    ,   bv  — 

AVe  multiply  by  in  by  multiplying]^  the  numerator  (Th.  T), 
and  we  divide  by  )i  by  multi])lying  the  denominator  (Th.  II). 


Hence  the  product  is 


am 

• 

bu 


That  is,  tJir  product  of  the  innvrrntnvfi  7.'?  ihc  nunirr- 
ator  of  tlic  rcf/uin'd  fnirfion,  (tnd  the  /nvditct  of  tlie 
denomlniitors  is  its  dcnoniinator. 


Multiply 

I. by  x  —  a. 

X  —  a     •' 

ab    , 

3-    -_^.  i>y  ^1/- 


EXERCISES. 

ab 


2. 


1        ^ 

by  -' 


A      by  ar*  —  c?, 

x  —  a     -' 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


V 


// 


Cp, 


y. 


Ua 


1.0 


1^ 


M 

2.2 


I.I 


ieJ  ^^^ 

I:      la^        ^ 

^    I-    12.0 


1.25 


U    1111.6 


Photographic 

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(716)  872-4503 


4^ 


V 


iV 


<s 


I 


r 


80 


ALGEBRAIC   OPERATIONS. 


ahn 


::  by  ^'f 


m 


by  ax^  -\- 


m  —  a 

X  —  m 


a  —  h  ,      a  +  b 

7.     by 


8.     a  A DY  n  A 

11     ■^  m 


-.      .X  ,  y—ah  m  -\-  n  ,      n  —  m 

o.     ah by  r/y  4- • 10.     by  — 

^  y  X  111  —  n    ''  in  +  u 

,,  ,,.  ,  bx  ,      a       1)      X 

II.     Multii)ly  a  -\ by  7  4-  -  A 

^  ■^  m     •'   b      X      a 


Ans.  — ■' 


12.  lieduce  \in  A \\m 1. 

V  m  —  111  \  m  +  nl 

13.  Reduce  («  —  -  )  U}  — j-\' 

14.  Multiply  b  ■—  -^  by  -• 

^  -^  a     -^  X 

15.  Divide  —  by  /?. 

16.  Divide ,  by  a  +  b. 

a  —  b    -^ 

1 7.  Divide by  x  —  1. 

X  -{-I     -^ 

18.  Divide  -—^^  by  1  +  x\ 

19.  Divide- — -— — -  —  by  i«  —  rt«. 

a"  -\-  b^ 

109.  Beciprocal  of  a  Fraction.    The  reciprocal  of 
a  fraction  is  formed  by  simply  inverting  its  terms. 

For,  let  T  >)e  the  fraction.     By  definition,  its  reciprocal 
will  be 

a 

Multiplying  both  terms  by  b,  the  numerator  will  be  h  and 

the  denominator  ,  x  b,  that  is,  a. 
b 

Hence  the  reciprocal  required  will  be     ,  or,  in  alirebraic 
1  a  ° 

language, 


FRACTIONS. 


m. 


a        a 


110.  Bif.     A  Complex  Fraction  is  one  of  wliich 
either  of  tlie  terms  is  itself  fmctional. 

a 
b 


Example. 


m  + 


X 

y 


IS  a  complex  fraction,  of  which  ,  is  the  numerator,  and  m  +  - 
the  denominator.  V 

Tlie  terms  of  the  h.^sser  fractions  wliich  (Miter  into  the 
numerator  and  denominator  of  tlie  main  fraction  may 
be  called  Minor  Terms. 

Thus,  h  and  y  are  minor  denominators,  and  a  and  x  are 
minor  numerators. 

To  reduce  a  complex  fraction  to  a  simple  one,  miilti- 
plij  both  terms  hij  a  multiple  of  tlie  minor  denominators. 

am 


Example.    Reduce 


?/ 


b      h 

-  +  - 

y     X 

Multiplying  both  terms  by  xif,  the  result  will  be 

amx 

bxij  -f-  ][}f^ 

wliich  is  a  simple  fraction. 

EXERCISES. 

Reduce  to  simple  fractions  : 


1  + 


X 


I. 


y 

1 

a 

X 

~~y 

—  X 

a 

+  x 

n 

-\-x 

a  + 


x 


2. 


a  — 


X 


4. 


a  —  x 
6 


ah 

711)1 

hm 


82 


ALGEBUAIU   OVER.  1  TlONti, 


II. 


13- 


1  + 


1} 


1  - 

am  + 


/*  +  1 

7i'— T 


6. 


w  -(-  1 


ail. — 


8. 


1  + 


n 


1  — 


^2  -  rt^ 
2ab 


10. 


w'^  +  -,  +  3 


r/.^ 


12. 


-  +  « 


a  +  ^Z* 


ft 


14. 


a  +  b 


1 

+  x 

1 



X 

1 

—  X 

+ 

i 

"+ 

X 

1 

+  :>' 

1 

— 

x 

1 

—  X 

2x  - 

3 

1 

+ 

• 

x 

a 

+  d 

— 

x 

1 

a 

1 

+  a 

+ 

I 

a 

1 

a 

1 

—  « 

1 

+  a 

^2 

1 

^3   + 

a 

• 

^ 

1 

1 

6 

b 

+ 

a 

X 

-y 

+ 

V  + 

X 

X 

+  // 

f 

— 

X? 

X 

+  // 

x^ 

— 

t 

X 

-y 

x^ 

— 

t 

Division  of  one  Fraction  by  Anotlier. 


a 


m 


111.  Let  us  divide  ^  by  —    The  result  will  be  expressed 

by  the  complex  fraction 

a 
J 

n 
Reducing  this  fraction  by  the  rule  of  §  110,  it  becomes 

an 
bm' 


which  is  equal  to 


a       n 
y  X  — - 
0      m 


That  is, 


To  (Uvide  hy  a  fraction,  we  have  only  to  multiply  hy 
its  reciprocal. 


FRACTIONS. 


83 


Divide 
ah 


I. 
3- 

8. 


a 


a-b^'U 


X 


X 


X 


EXERCISES. 


x  —  1^  xi—l 


X  -\-\  ,      2x 
2.    -g-byy 


a'  -  lA 


a 
b 


^y 


a^  +  nb 


a 


,      a      m  ,     b       n 
6.     rr  -\ by . 


a      m 


a       be.      7)1       71      p 

+  -  +  -  by  —  +  -  +  ^  • 
"J      z    ''    X       y       z 


X 


a 


z     "     X 

b 


a 


b} 


r  + 


a 


a  +  b    ''  a—  I)      a  +  b 


Reciprocal   Relations    of  Miiliiplicatioii   and 

Division, 

113.  The  fuiidamental  principles  of  the  operations  npon 
fractions  are  included  in  tlie  following  summary,  the  under- 
standing of  which  will  afford  the  student  a  test  of  his  grasp  of 
the  subject. 

1.  The  reciprocal  of  the  reci})rocal  of  a  number  is 
equal  to  the  number  itself.    In  the  language  of  Algebra, 

J_ 
T 
a 


=  a. 


2.  The  reciprocal  of  a  monomial  may  be  expressed 
by  chauginp:  the  algebraic  sign  of  its  exponent. 

3.  To  multiply  by  a  number  is  equivalent  to  dividing 
"by  iia  veciiproc'dl,  vnd  vice  versa.    That  is, 


I^ 


and  vice  versa, 


a    or    — -  =  aiV", 
a 

iV  X  -  =  —  • 
a        a 


■  4 

'ill 


84 


ALGEBRA  IC    OPEU.  1 TIONS. 


4.  When  tlie  numerator  or  denominator  of  a  fraction 
is  a  product  of  H(n<'ral  factors,  any  of  tlies(»  factoi's  may 
1)0  transferred  from  on<;  term  of  the  fraction  to  the  other 
by  changing  it  to  its  reciprocah     That  is, 


ttbc 

V<1'' 


he 


_  2^ 


ahc 


a 


Or, 


pqr 
he 


qr 


etc. 


etc. 


nbc he     p~k(hc 

pqr       (r^pqr  rjr    ' 

5.  JfuUfplleation  hy  a  ftictor 

gr(?ater  than  unity  increases y 
h^ss  tlian  unity  diminishes. 
Division  by  a  divisor 

greater  than  unity  diminisltes, 
less  than  unity  increases. 

6.  («)  Wlien  a  factor  becomes  zero,  the  product  also 
becomes  zero. 

{ft)  AVhen  a  denominator  becomes  zero,  the  product 
becomes  infinite.    That  is, 

a 


0 


=  infinitv. 


N"oTE.  The  folloAV'iiig  way  of  expressing  what  is  meant  by 
this  last  statement  is  less  simple,  but  is  logically  more  correct: 

If  a  fraction  has  a  fixed  numerator,  no  matter  how 
small,  we  can  make  the  detiominator  so  much  smaller 
that  the  fraction  shall.be  greater  than  any  quantity  we 
choose  to  assign. 

EXERCISE. 

If  the  numerator  of  a  fraction  is  '2.  how  small  must  the 
denominator  1)0  in  order  that  the  fraction  nuiy  exceed  one 
thousand?  That  it  may  exceed  one  mihion  ?  That  it  may 
exceed  one  thousand  millions? 


'{ 


< . 


BOOK    III. 
OF    EQUA  TIONS. 


CHAPTER    I. 
THE     REDUCTION     OF     EQUATIONS. 


Definitions. 

113.  Def.  An  Equation  is  a  statement,  in  the  lan- 
gTiag<.^  of  Algebra,  tliat  two  exx)ressions  are  equal. 

114.  Def.  The  two  equal  expressions  are  called 
Members  of  the  equation. 

11.3.  Def.  An  Identical  Equation  is  one  which  is 
tnie  for  all  values  c»f  the  algebraic  symbols  wliicli  enter 
into  it,  or  which  has  numbers  only  for  its  members. 

Examples.    The  equations 

14  +  0  =  29  —  G, 
5  +  13  —  3  X  4  —  G  =  0, 
which  contain  no  algehraic  symbols,  are  identical  equations. 
So  also  are  tlie  equations 

X  -m  X, 

X  —  X  —  0. 
{x  +  ({)  {x  —  a)  =  x^  —  (fi, 

a  +  y)(i-^)-i  +  y/-o, 

because  tliey  are  necessarily  true,  whatever  values  we  assign  to 
X.  a.  and  ?/• 

Rem.  Ail  the  e((uations  used  in  the  preceding  two  hooks 
to  express  the  relations  of  algebraic  quantities  arc  identical 
one?,  because  they  are  true  for  all  values  of  thcsj  quantities. 


80 


EQUATIONS. 


1 1(5.  Dt'f.  All  Equation  of  Condition  is  one  wliicli 
cfin  1)(3  true  only  wlien  tlic  Jilgcbiaic  symbols  Jire  cciiuil 
to  certain  quantities,  or  have  certain  relations  among 
themselves. 

Examples.     Tlu  equation 

a:  4-  C  =  22 

can  bo  true  only  when  x  is  equal  to  IG,  and  is  therefore  an 

equation  of  condition. 

The  equation 

X  -\-  h  =  a 

can  be  true  only  Avlien  x  is  equal  to  the  difference  of  the  two 
quantities  a  and  h. 

Rem.  In  an  equation  of  conditivm,  some  of  tbe  quantities 
may  be  supi)osed  to  be  known  and  others  to  be  unknown. 

lit.  Def.  To  Solve  an  equation  means  to  lind 
some  number  or  algebraic  expression  which,  being  sub- 
stituted for  the  unknown  quantity,  will  render  the 
equation  identically  true. 

This  value  of  the  unknown  quantity  is  called  a  Root 
of  the  equation. 

EXAM  PLE  S. 

1.  The  number  3  is  a  root  of  the  equation 

?:.7,3  —  18  =  0, 

because  when  we  put  3  in  jdace  of  x,  the  equation  is  satisfied 
identically. 

2.  The  expression is  a  root  of  the  cquntion 

'zcx  —  4rt  +  2Z>  =  0, 

when  X  is  the  unknoAvn  quantity,  l)ecausc  when  we  substitute 
tli's  expression  in  place  of  ;r,  we  have 

<1n  —  b\ 


2c 


or  4rt  —  2b 

which  is  identically  true. 


(ll.Zl^)  _  4«  +  2^-  =  0, 
4a  4-  2&  =  0, 


one 


AXIOMS. 


87 


Rkm.  It  is  common  in  Elcmctitury  Alc;ol)ni  to  iv}>rcsi'nt 
unknown  (luantities  l)y  tlic  last  letters  of  the  til})li!il)et,  luid 
(juantities  snppused  to  bo  known  by  the  llrst  letters.  Uut  this 
is  not  at  all  necessary,  and  the  student  should  accustom  hini- 
scK  to  regard  any  (va  symbol  as  an  unknown  (juautity. 

Axioms. 

118,  Def.  An  Axiom  is  a  proposition  wliicli  is 
talien  for  granted,  without  proof. 

Equations  arc  solved  by  o}ierations  founded  upon  the  fol- 
lowing axioms,  which  arc  seli'-evident,  and  so  need  no  prouf. 

Ax.  I.  If  equal  quantities  be  added  to  the  two 
members  of  an  equation,  the  members  will  st'll  be  equal. 

Ax.  II.  If  equal  quantities  be  subtracted  from  the 
two  members  of  aa  equation,  they  will  still  be  equal. 

Ax.  IIx.  If  the  tvro  members  be  multiplied  by  equal 
factors,  they  will  still  be  equal. 

Ax.  IV.  If  the  two  members  be  divided  by  equal 
divisors  (the  divisors  being  difierent  from  zero),  they 
will  still  be  equal. 

Ax,  V.    Similar  roots  of  the  two  members  are  equal. 

Tlieso  axioms  may  be  summed  up  in  the  single  one, 

liiiiiilar  opcrtdioiis  upon  equal  quantities  i^ive  equal 
results. 

IH).  An  algebraic  equation  is  solved  by  performing 
uch  similar  operations  upon  its  two  members  that  i\\Q 
unknown  qujuitity  shall  finally  stand  alone  as  one 
member  of  an  equation. 

Operations  of  Addiiioii  and  SiLbtractioii — Trans- 

posinj]^  Terms. 

130.  Theorem.  Any  tenn  may  be  transposed  from 
one  member  of  an  equation  to  the  other  member,  if  its 
sign  be  changed. 


'    I 


.  ^..1.: 


I 


u 


88 


EQUATIOXH. 


Proof.     Let  us  put,  in  nccordanrc  willi  g  W,  2cl  Prin., 

/,  uiiy  term  of  eitluT  nit'iulrjr  of  the  c((utilioii. 

((,  nil  the  olluT  terms  of  the  same  member. 

b,  tlie  opposite  member. 

The  ccpuuion  iis  then 

a  -{-  i  —  b. 

Now  su])tract  /  from  both  sides  (Axiom  II), 
«  +  /  —  /  =  />  —  /; 

or  by  reduction,  a  =  h  —  i. 

This  e(|Uii(ion  is  the  same  us  tjje  one  from  wliieli  we  started, 
cxcei)t  tb;it  /  has  been  transposed  to  tbu  second  member,  with 
its  si<:fii  (•lKinp;ed  fi-om  +  to  — . 

If  (he  equation  is 

b  —  i  =  n, 

we  may  add  /  to  both  members,  wliieh  woukl  givo 

b  =  a  -{-  f. 

NUMERICAL      EXAMPLE. 

The  learner  will  test  each  side  of  the  following  equations  : 

19  +  3_9  +  4  =  7  +  10. 
10  +  3  —  0  —  7  +  10—4. 
19  +  ;i  =  7  +  10—4  +  9. 

3  =  7  +  10—4  +  0  —  19. 
0  —  7  +  10—4  +  0  —  19—3. 

131.  Rem.  All  the  tonus  of  either  member  of  an 
equation  may  bo  transposed  to  the  other  member, 
leaving  only  0  on  one  side. 

Example.    If  in  the  equation 

b  =  a  +  f, 
we  transpose  b,  we  have    0  =:  a  -\-  t  —  b. 

By  transi)osing  a  and  /,  we  have 

b  —  a  —  t  =  0. 

133.  CJ/atif/uif/  Sif/us  of  Mt'Dibcrs.  If  we  change  tlie  signs 
of  all  the  terms  in  both  members  of  an  equation,  it  will  still 
be  true.    The  result  will   be  the  same  as  multiplying  both 


Transposing  4, 
9, 
19, 
3, 


it 
it 

u 


or 


REDUCTION. 


80 


mcml)C'r.s  by  —  1    )r  triiiispo.-in^  all  the  terms  of  each  nii'iiibcr 
to  tliu  oLlior  side,  uiid  thtii  cxclmiiging  lliu  lermd. 

Exami'm:.     The  c((iiati()n 

17  -f  8  =  11  +  11 

may  Ik-  traiislormed  into      0  =  11  +  II:  —  17  _  8, 

or,  0  =  —  11  —  11  +  ir  -f-  8, 


or, 


17  — 8  =  —  li_U. 


Operation  of  MiiIlix>li<'utiGii. 

12J?.  Charing  of  Frarfiona.  T]io  opcrsition  of  muKipll- 
cation  is  usually  pcrforuied  u{)i)ii  thy  two  sides  of  au  c<|uati()ii, 
iu  order  to  clear  the  e([uatiou  of  fractious. 

To  clear  an  eciuatiou  of  fractious: 

First  Mirnroi).  .Mitltiplii  iU  Dtmihrrs  hy  the  least 
comniofb  initUi])le  of  till  Us  ilciioniiiidtoi's. 

Sk(;ond  Mktiiod,  .MHIfinhf  its  mcnihrrs  hij  each  of 
tha  deiu)i)ii]i(tiors  in  succession. 

Rem.  1.  Sometimes  the  one  and  sometimes  tlic  otluT  of 
these  methods  is  the  uioro  conveuieut. 

Rhm.  '^.  The  operalion  of  cleariug  of  fractious  is  similar 
to  that  of  reducing  fractions  to  a  commou  deiu»iniuator. 

Example  of  Fikst  Method.  Clear  from  fraclious  the 
e(iuatiou 

4  +  U  +  8  -  ^^- 

Hero  24  is  the  least  common  multiple  of  the  denominators. 
Multiplying  each  term  by  it,  \vc  have, 

Qx+  Ax-\-  3.^•  =  G2-1, 
or  l^x  =  62-i. 

Example  of  Second  Method.     Clear  the  equation 
a 

X 


a 


n  c 

X  -f-  a       X 


0. 


Multiplying  by  x  —  a,  we  find 


ax 
a  4- 


X  -\-  a 


+ 


ex 


ea 


X 


=  0. 


l^ 


\^=r 


a  I 


'9 


00  EiiUATiONii. 

Multiplyinpf  by  x  -\-  a^ 


ax  -f-  a^  +  ^^r  —  rt'  + 


X 


=  0. 


lit'tlucing  ;iiul  miillijilying  by  x, 

)lax^  +  cx^  —  ca^  =  0. 

EXERCISES. 

Clear  (lie  following  C([iiiiLions  ol'  IVuctions  : 


I. 

2.r 

2. 

a;      X 
5  ""7 

=  70. 

3' 

a:       .r       a:  _ 

4- 

a:      a:8 

b 

"    '         • 

If- 

5- 

^  4.:'/  +  ^  ==.!.. 

6. 

a       h 

3  +  4 

X 

~  5' 

7. 

r»              a;      _ 

8. 

a* 
a;  —  a 

2x 

X  —  a      X  i-  a  ~ 

~  X  +  b 

9- 

X  -{■  ff       .7;3  -j-  2nx 
X  —  a          X  —  a 

10. 

X  —  2 

x-b 

X  +  2 

""  a:  4-  5 

11. 

X      y  _  a 
y~  x~  V 

12. 

X  ~  a 
X  +  a 

a;  4-  rt      a; 
x  —  a      a 

13- 

X       ,       y 

«  —  6       6  —  rt 

Hero  tLe  second  term  is  the  same  as  — 

a 

-y 

-b 

1  A. 

a;  4-  a       a-  —  i 

. .          ' • 

=  0. 


a  —  a; 


a;  —  a 


Reduction  to  the  Xormjil  Form. 

124.  Dcf.    An  equation  is   in  its  Normal  Form 

when  its  terms  are  reduced  and  arranged  according  to 
the  powers  of  the  unknown  quantity. 

In  tbc  normal  form  one  mcml)C'r  of  the  ecjuution  is  expressed 
as  tin  entire  function  of  the  unknown  <jiiantity,  and  the  other 
is  zero.     (Compare  §§  50,  70.) 

To  reduce  an  ecpiation  to  tlie  normal  form: 
I.  Transpose  all  the  terms  to  one  mnnhcr  of  the  equa- 
tion, so  as  to  leave  0  as  tlie  other  nieinher. 


liklDlX'VlON. 


01 


IT.  Clear  the  r qua t ion  nf  fvacthnfi. 

III.  I'lcar  the  ('(/Hftfion  of  pftrcnt/icsrs  hij  prrfoi'unit'i 
(ill  llic  ()/)(r(tfions  intlicafcit. 

IV.  Collect  vuclh  stf  of  firms  ronhfininjj  li/\:c  poH'cr^f 
of  the  itnkiioii'ii  (jmiiifilij  info  <i  sinijlc  our. 

V.  Dirhir  hij  nin/  cinunioii  fdctur  wkicli  docs  nob  con- 
tain the  unknoirn  (jnantitn. 

T?i:m.  This  ordrr  of  opernlions  may  be  deviated  (Vom 
ac'C'onling  tocircimistanci'S.  Aftcra  little  j)raetiee,  tlieKtiident 
may  take  the  shortest  way  of  reaching  the  re.'jult,  without  re- 
spect to  rules. 

EXAMPLES. 

I.  lleduce  io  the  normal  rurm 

{x  -  2)  (x  -  3)  ^  {x  +  2)  (x  +  A) 

X  —  6  X  +  xi 

1.  Clearing  of  fractions, 

(x  +  5)  {x  -  2)  {X  -  ;5)  =  {x  -  5)  {x  f  2)  {x  +  4). 

2.  Performing  the  indicated  operations, 

a;3  —  VJx  +  30  =  x^  -f-  x^  —  ^2.^  —  40. 

3.  Transposi\»g  all  the  terms  to  the  second  member  and 
reducing,  \ 

0  =  .r2  -  3x  —  70, 

which  is  the  normal  form  of  the  equation. 

Rem.  Had  we  transposed  the  terms  of  the  second  member 
to  the  lirst  one,  the  result  would  have  been 

—  x^  +  3x  -j-  TO  =:  0. 

Either  form  of  the  equation  is  correct,  but,  for  the  sake  of 
uniformity,  it  is  customary  to  transpose  the  terms  so  that  the 
coelUcient  of  the  highest  power  of  x  shall  be  positive.  If  it 
comes  out  negative,  it  is  only  necessary  to  change  the  signs  of 
all  the  terms  of  the  equation. 

Ex.  2.     lleduce  to  the  nornud  form, 


bmx^ 


X 


a 


2  fix  duix^         „  _ 

o -«  =  2mx  —  int. 

X  -{-  ti       x^  —  «■* 


92 


EQUATIONS. 


1^ 


1 


1.  Transposing  to  the  first  member, 

:, 5  —  2mx  -\-  6a  =:  0. 

X  —  a      X  -\-  a      "'-      "^ 


it?  —  d^ 

2.  To  ck'iir  of  fractions,  we  notice  that  the  least  common 
multii)lo  of  the  denominators  is  x-  —  a^.  Multiplying  each 
term  by  this  i'actor,  Ave  have, 

5mx'^  {x+a)—2ax{x—a)—3mx^—2uix{x^—a''^)-{-ba{x^—a')  =  0. 

3.  Performing  the  indicated  operations, 

bmx^ + ixcinx^ —2ax^  +  2ci?x — 3inx^ —2mx^ + 2cfimx + bax^—ba^=0, 

4.  Collecting  like  powers  of  .r,  as  in  §  7G, 

(3a  +  bam)  x^  +  (2rt2  +  2a~rn)  x  —  5«3  =  0. 

5.  Every  term  of  the  eqnation  contains  the  factor  a.  By 
Axiom  IV,  §  118,  if  both  memlters  of  the  equation  be  divided 
by  a,  the  ecpiation  Avill  still  be  true.  The  second  member 
being  zero,  Avill  remain  zero  "when  divided  by  a.  Dividing 
both  members,  we  have 

^^3  -1-  hm)  x^  +  2a  (1  +  m)  x  —  ba^  =  0, 

Avhich  is  the  normal  form. 


*  > 


EXERCISES. 

Eeducc  the  following  equations  to  the  normal  form,  x,  y, 
or  ;;  being  the  unknown  quantity : 

X  —  a  _x  -\-  a 
A,  '    X  -\-  a  ~      X 

x  —  '7         2x-\-6 


3f  +  2ij  _  71- J^ 


3- 
4. 


6. 
7- 


2x  +  10        4x  —  2 
a;3  _  SaJ^x  +  2«3 


2x  -\-  a 


i^^  —  bax^ 

x^—  bax  =  --T • 

2x  —  a 


5-    ~'^.  +  --^^  +  ;:&  =  ^• 


+ 


a 


+ 


a  i-  b       h  -\-  z       a  +  z 

«8 


-  =  0. 


+ 


Z^ 


ah 


a  —  z   '  a^  —  x^ 


z^  —  a^ 


i\ 


REDUCTION. 


93 


8. 


10. 


II. 


13- 


14. 


15. 


16. 


^       G       5        4        ^ 

7  + +    ,  +  -3  =  0. 


a;  — 

a 

+ 

x^- 

cfi 

4 

5 

z 

-1- 

¥ 

^ 

c  — 

■2 . 

;;' 

r^ 

= 

b 

X  — 

—  • 

a 

12. 

6- 

1 

X 

« 

+ 

a^ 

«3 

""  a;3* 

a  — 

1 

cfi- 

1 

x^ 

32! 

2  + 

T 
2 

— 

5^2 

3z  — 

■3 

z 

1 

'    -~  • 

2 

ai 

1 

1 

hx 

1  - 

X 

1 

+ 

a 

1 

1 

it  — 

« 

a 

a 

h 

a 

X 

i  — 

-  X 

- 1. 


J6 


b 

X 


a  — 


m 


m 


nx 


71 


X 


x  + 


X 


'I: 


% 


Degree  of  Equations. 

135.  I)ef.  An  equation  is  said  to  be  of  tlie  n*^  de- 
gree wlien  n  is  the  highest  power  of  the  unknown 
quantity  wliicli  appears  in  the  equation  after  it  is  re- 
duced to  the  normal  form. 


EXAMPLES, 

The  equation      Ax  -\-  B  =  ^      is  of  the  first  degree. 

Ax^  -^  B  —  ^      "       "  second     " 

Ax^  j^Bx-\-  C  —^      "       "  tliird       « 
etc.  etc. 

An  equation  of  the  second  degree  is  also  called  a 
Quadratic  Equation. 


!     f 


94 


EQUATIONS  OF  THE  FlllST  DEGREE. 


An  oqiiation  of  the  third  degree  is  also  called  a 
Cubic  Equation. 

Example.     Tlie  o(iu:ition 

ax^  +  hx^if  +  y^  +  n^^"  —  0 
is  a  quadratic  equation  in  x,  because  oi?  is  of  the  highest  power 
of  .r  which  enters  into  it. 

It  is  a  cubic  equation  in  y. 

It  is  of  tlic  first  degree  in  z. 


-♦►♦>- 


'»'!    1 


'% 


CHAPTER    II. 

EQUATIONS    OF    THE    FIRST    DEGREE    WITH    ONE 
UNKNOWN    QUANTITY. 

12(5.  Remark.  By  the  preceding  definition  of  the  degree 
of  an  equation,  it  will  be  seen  that  x\\  equation  of  the  first 
degree,  with  x  as  the  quantity  supposed  to  be  unknown,  is  one 
which  can  be  reduced  to  the  form 

lix  +  D  =  <),  (a) 

A  and  B  being  any  numbers  or  algebraic  expressions  wliich 
do  not  contain  x. 

Such  an  equation  is  f'-'^quently  called  a  Simple  Equation. 

Solution  of  Equations  of  the  First  Degree. 

127.  If,  in  the  above  equation,  we  transpose  the  terra  B 
to  the  second  member,  we  have 

Ax  =  —B. 

If  we  divide  both  members  by  A  (§  118,  Ax.  IV),  we  have, 

B 

• 

A 

Here  we  have  attained  our  object  of  so  transforming  the 
equation  that  one  member  shall  consist  of  x  alone,  and  the 
other  member  shall  not  contain  x. 


x  = 


i^ 


ONE    UNKNOWN   QUANTITY. 


B 


95 


To  prove  that  — -.  is  the  required  vahie  of  .r,  we  sub.sti- 
tute  it  for  x  in  the  equation  («).     The  equation  then  becomes, 


-'^-  +  i;  =  C; 


or,  by  reducing, 


-i?  +  B 


0, 


an   equation   which  is    identically  true.     Therefore, r  is 

the  required  root  of  the  equation  («).     (§  117,  Dvf^ 

138.  In  an  equation  of  the  first  degree,  it  will  be  unnecc??- 
sary  to  reduce  the  equation  entirely  to  the  normal  form  Ijy 
transposing  all  the  terms  to  one  member.  It  will  generally  1)0 
more  convenient  to  place  the  terms  Avhich  do  not  contain  x  in 
the  opposite  member  from  those  which  are  multiplied  by  it. 

Example.    Let  the  equation  be 

mx  +  rt  =  nx  -\-  1).  (1) 

TVe  may  begin  by  transposing  a  to  the  second  member  and 
nx  to  the  first,  giving  at  once, 

mx  —  nx  r=z  1)  —  a, 

or  {m  —  n)  x  =  h  —  a, 

without  reducing  to  the  normal  form.  The  final  result  is  the 
eame,  Avhatever  course  we  adopt,  and  the  division  of  both 
members  by  vi  —  n  gives 


X  = 


h  —  a 


III 


n 


139.  The  rule  which  may  be  followed  in  solving  equations 
of  the  first  degree  with  one  unknown  quantity  is  this : 

I.  Clear  the  equation  of  fractions. 

II.  Transpose  the  terms  irh'ich  are  mnlti plied  hy  the 
unlcnoiun  qiuintity  to  one  member ;  those  ichich  do  not 
contain  it  to  the  other. 

III.  Divide  hy  the  total  coefficient  of  the  unknown 
quantity. 


C  M 


«■; 


m 


96 


EQUATIONS   or   TllK  FimT  DEGliPJE. 


NoTK.  Rules  in  Algobra  arn  given  only  to  enable  the  beginner  to  go 
to  work  in  a  way  which  will  always  be  Huie,  thougii  it  nniy  not  always 
be  the  ahortest.  In  solving  equations,  be  should  emancipate  himself 
fiom  the  rules  as  soon  as  i)ossible,  and  be  prepared  to  solve  each  ecjua- 
tion  jjresented  by  such  jjrocess  us  apjxsars  most  concise  and  elegant.  No 
ojx'ration  ui)on  the  two  nu-mbers  in  accordanci?  with  tlie  axioms  (^  118) 
can  lead  to  incorrect  results  (provided  that  no  <iuuntity  which  becomes 
zero  is  used  as  a  multiplier  or  divisor),  and  the  student  is  therefore  free 
to  operate  at  his  own  pleasure  ou  every  equation  presented.  » 


":;  I- 


"r 


I.  Given 


EXAMPLES. 

ax 


hil 


=  1. 


It  is  required  to  find  the  value  of  eacli  of  tlie  quantities  a, 
h,  X,  and  //,  in  terms  of  the  others. 
Clearing  of  fractions,  Ave  have 

ax  =  I))/. 
To  find  a,  we  divide  by  .  ,  which  gives 

In/ 

X 

To  find  b,  we  divide  by  y,  which  gives 

ax 


y 


=  h. 


To  find  x,  we  divide  by  a,  which  gives 

hii 
a:  =  —  • 
a 

To  find  y,  we  divide  by  h,  which  gives 

ax 

Thus,  when  any  three  of  the  four  quantities  a,  l,  x,  and  y, 
are  given,  the  fourth  can  be  found. 

2.  Let  us  take  the  equation, 

x  —  'i    _  2x  +  6 
2a:  +  10  ~  4.«  —  2 

Clearing  of  fractions,  we  have 

4.f2  —  30.1-  +  14  =  4a;'2  +  32x  +  GO. 


ONE    UNKNOWN  QUANTITY 

Transposing  and  reducing, 

—  ij-Zx  =z  4G. 

Dividing  both  members  by  —  G2, 

40  _  _  40  _  _  23 

—  G2  ""       \}2  ~  ~  3l' 


97 


This  result  should  now  be  proved  by  computing  the  value  of  both 


2:}  . 


members  of  the  original  e(iuation  when  —  ["  -  is  substituted  for  a*. 

ol 

X       X       ax       1 
m      n        b       m 

Proceeding  in  the  regular  way,  we  clear  of  fractions  l)y 
multii)lying  by  m)ih.     This  gives 

nbx  4-  mbx  =  amnx  —  nb. 

Transposing  and  reducing, 

{nb  +  mb  —  amn)  x  •=  —  nb. 

Dividing  by  the  coefficient  of  x, 

_  nb  _  nb 

~       nb  +  mb  —  amn  ~  amn  —  mb  —  nb 

These  tAvo  values  are  equivalent  forms  (§  100). 

But  we  can  obtain  a  solution  without  clearing  of  fractions. 
ax 


Transposing  — ,  we  have 

X       X      ax 1 

m       71        b  m' 

which  may  be  expressed  in  the  form 

/I       1       a\               1 
(-  H j)x  = 

\m      n       bl  m 

Dividing  by  the  coefficient  of  x, 

m 


1  j_  1  _? 
m      71      b 


This  expression  can  be  reduced  to  the  other  by  §  110. 
1 


98 


EQUATIONS  OF  THE  FIRbT  DEGREE. 


\i      i 


(P  i 


k    a. 


EXERCISES. 

Fiml  the  values  of  x,  //,  or  u  in  the  following  cqiuitioiis: 
5_—  '.\x  _  8,/'  —  \) 


3 
5 

7 

9 
II 

12 

13 
14 

15 
16 

17 
18 

19 
20 


3 


re      a;      ^  _  90 
a       b      c 


2. 


.'•  =^  a. 


4.     =-  =  9. 

X  —  \ 

6      ~~  —  — 
V.  —h  ~  u 


n      u      u 

3  —  ^  +  p  =  ?.  -  2G.       8.     a  —  Ix  =  h  +  ax. 


u       7t    _1       1 
7t'^  b  ~  7i^  b 


n      ,  3  —  .r 

10.  ox    H ;~-    =    X. 

o 


a      _      c 
c  —  X.       a  —  X 

X—  l_rr  —  3_r?—  5       a-  —  G 

x  —  'z     x  —  ',^~x~{j~  X  —  Y 

—  jj  7:=  a  —  h. 

_J 1_  ^      1 1_ 

X  —  'Z      X  —  4:~  X  —  yj      X  —  8* 


I — 


a 


a      b  —  a       b  -{-  a 

-.  X        1 

ax  +  6  =  -  +  y 
rt       b 

!ilL^  _l_  ^^  ~  ^    ,   ^^  —  g  _  ?^  —  (rr  4-  J  +  c) 

^  c  «      ~  7ihc 

m  [x  +  «)    ,   n  (x  +  Z») 

~x-TT-  +  -tT'^T"  =^  ^«  +  ''- 

(:,_«)3  +-(.,_^i)3  +  (.,._^)3  ^  3  (..,_^)  (,_j)  ^^,_^^^ 

Find  the  values  of  each  of  the  four  quantities,  a,  b,  c,  and 
d,  in  terms  of  the  other  three,  from  the  eq^'ations 


21. 


(^       ,       (I  r,  f'b      , 

+   / 77  =  0.  22.         -.  +  1   =  0. 


Z>  —  c       b  —  d 


ONE    UNKNOWN  QUANTITY. 


99 


i'robloms  le'»'liiijjf  to  Siinplo  Eciiiatioiis. 

loO,  Tlie  fir,;t.  (liilieulty  wliicli  tlio  boirinncr  meets  witli  in 
the  s<.»lu lion  of  an  algebraic  ])roblein  is  to  st;ite  it  in  the  tbriii 
of  an  equation.  This  is  a  i)roecss  in  whieh  the  student  must 
dt'iiend  upon  his  own  powers.  The  following  is  the  general 
pliiii  of  jtroeeeding  : 

1.  Study  tne  pro])lcm,  to  ascertain  what  quantities  in  it 
are  unknown.  There  may  be  several  such  quantities,  but  the 
problems  of  the  present  chapter  are  such  that  all  these  quan- 
tities  Ciiu  be  expressed  in  terms  of  some  one  of  them.  Scleeti 
that  one  by  whieh  this  can  be  most  easily  done  as  the  unknown 
quamity. 

'Z.  Represent  this  unknown  quantity  by  any  algebraic  .^^ym- 
bol  whatever. 

It  xs,  common  to  select  one  of  the  last  letters  of  the  alpha- 
bet for  the  symliol,  but  the  student  should  accustom  himself 
to  work  equally  well  with  any  symbol. 

3.  PerfoF!  on  and  with  these  symbols  the  operations  rc- 
qninxl  by  the  problem.  These  ope-rations  are  the  same  that 
wouW  l>e  necessary  to  verify  the  adopted  value  of  the  unknown 
finanlity. 

4-  Express  the  conditions  stated  or  implied  in  the  problem 
by  means  of  an  ecpiation. 

5.  The  solution  of  this  equation  by  the  methods  already 
explained  will  give  the  value  of  the  unknown  (pumtity.  It  is 
always  Ix^st  to  verify  the  value  found  for  the  unknown  (puin- 
tiiy  by  operating  upon  it  as  described  in  the  equation. 


\ 


-c). 
and 


EXAMPLES. 

I.  A  sum  of  440  dollars  is  to  Ije  divided  among  three  people 
so  that  tlie  share  of  the  second  shall  be  30  dollars  more  than 
that  of  the  first,  and  the  share  of  the  third  SO  dollars  less  than 
those  of  the  first  and  second  together.  What  is  the  share  of 
each? 

SOLCTiox.  1.  Iloro  thoro  are  really  throe  unknown  qnantitios,  l)iit 
it  is  only  necessary  to  represent  the  share  of  the  lirat  by  an  unknown 


i       ;  M 


100 


EQUATIONS   OF  TUE  FIRST  DEGREE. 


2    Therefore  let  U3  put 

X  =z  share-  of  the  first. 

3.  Then,  by  the  terJis  of  the  atatenient,  the  share  of  the  second  will  bo 

X  +  ;}o. 

To  find  the  share  of  the  third  we  add  these  two  together,  which  makes 

2x  +  30. 
Subtracting  80,  we  have 

2x  —  50 
as  the  share  if  the  third. 

"We  now  add  the  throe  shares  together,  thus, 

Share  of  first,  x 

"  seeoml,      x  +  30 
"   third,       2rc^-J)0 

Shares  of  all,  ix  —  20 

4.  By  the  conditions  of  the  problem,  these  three  shares  must  together 
make  up  4-10  dollars.  Expressing  this  in  the  form  of  an  equation,  we 
have 

ix  —  20  =  440. 

5.  Solving,  we  And 

X  =:  llij  —  sliare  of  first. 

Whence,  115  +  30  =  145  =  sliare  of  second. 

115  +  145  —  80  =180  =  share  of  third. 

Sum  --  440.    Proof. 

Ex.  2.  Divide  tlie  number  00  into  four  parts,  such  tliat 
the  lirst  increased  by  2,  the  second  diminished  by  2,  the  third 
multiplied  by  2,  and  the  fourth  divided  by  2,  shall  all  be  ccpuil 
to  the  same  quantity. 

Here  there  are  really  five  unknown  quantities,  namely,  the  four  parts 
and  the  quantity  to  which  they  are  all  to  be  equal  when  the  o])eration  of 
adding  to,  subtracting,  etc.,  is  performed  upon  them.  It  will  be  most 
convenient  to  take  this  last  as  the  unknown  quantity.  Let  us  therefore 
put  it  equal  to  u.     Then, 

Since  the  first  part  increased  by  3  must  be  equal  to  u,  its  value  will 
he  u  —  2. 

Since  the  second  part  diminished  by  2  must  be  equal  to  u,  its  value 
w!n  be  u  +  3. 


Since  the  third  part  multiplied  by 


^  must  be  u,  its  value  will  be 
Since  the  fourth  part  divided  by  3  must  make  a,  its  value  will  be  2w. 


3 


ONE    UNKNOWN  QUANTITY. 


101 


Adding  these  four  parts  up,  their  sum  is  found  to  bo       • 

Wy  the  conditions  of  the  problem,  this  sum  must  make  up  the  num- 
ber DO.     Therefore  wo  have 

5}   -  -^w- 


Solving  this  equation,  we  find 

U 

Therefore 


20. 


1st  part  z=  u  —  2  =  \^. 
2a  ''  =  u  4  2  =  22. 
3d     ''     =:?^-i-2  =  10. 


4ti 


<; 


'hi        =  40. 


The  sum  of  the  four  e([ualsOO  as  required,  and  the  first  part  increased 
by  2,  the  secoud  diminished  by  3,  etc.,  all  malio  the  number  30,  as  re- 
quired. 


PROBLEMS     FOR    EXERCISE. 

1.  What  number  is  tliat  from  whicli  avc  obtain  the  pnmc 
resiilL  whether  we  multiply  it  by  4  or  subtract  it  from  100? 

2.  Whiit  number  is  that  which  <(ivcs  the  same  result  when 
we  divide  it  by  8  as  when  we  subtract  it  from  81  ? 

3.  Divide  284  dollars  among  two  people  so  that  the  share 
of  tlie  first  shall  be  three  times  that  of  the  second  and  $10 
more. 

4.  Find  a  number  such  that  \  of  it  shall  exceed  \  of  it 
by  12. 

5.  A  shepherd  describes  the  number  of  his  sheep  by  saying 
that  if  he  had  10  sheep  more,  and  sold  them  lor  5  dollars  each, 
he  would  have  G  times  as  many  dollars  as  he  now  has  sheep. 
How  many  sheep  has  he  ? 

6.  An  applewoman  bought  a  number  of  apples,  of  which 
00  proved  to  be  rotten.  She  sold  the  remainder  at  the  rate  of 
2  for  3  cents,  and  found  that  they  averaged  her  one  cent  each 
for  the  whole.     IIow  many  had  slie  at  first? 

7.  If  you  divide  my  age  10  years  hence  by  my  age  20  years 
ago,  you  will  get  the  same  tjuotlent  as  if  you  should  divide  my 
present  age  by  my  age  26  years  ago.     What  is  my  present  age? 

8.  Divide  $500  among  A,  B,  and  C,  so  that  ]3^*hTlTrTiave  T 
$20  less  than  A,  and  C  $20  more  than  A  and  B  twfietlmv-'    "*  " '' 


102 


EQUATIONS  OF   TUN   FlIiST  DlXillEE. 


|:i 


'y 


9.  A  fadicr  left,  JjflOOOO  lo  bo  dividcvl  nmon^u:  his  five  chil- 
dren, (liivc'tiii<,Mliii(  oiicli  should  receive  %')0K)  iiKtre  than  the 
uext  yoiin^'er  one.      What  was  the  .share  of  each  'i 

10.  A  man  is  f)  y.-ars  ol(h'r  than  his  wife.  After  tliey  havo 
hien  niani('<l  VI  years,  8  times  her  aue  wuuld  make  '\  timen 
his  ULje.     What  was  their  age  when  m- ' -dv 

11.  Of  three  hrotiiers,  the  youiiL,  '  .  8  years  younger  than 
the  second,  and  tJie  ehk'st  is  as  okl  as  the  otiier  two  together. 
In  10  years  the  sum  ol'  their  ages  will  he  VIK).  What  are  their 
present  agesV 

II.  'rh(>  head  of  a  fish  is  0  inches  long,  the  tail  is  as  long 
as  the  head  and  half  the  hody,  and  the  hody  is  as  long  as  the 
Jiead  and  tail  together.     What  is  the  whole  length  of  the  (Ish? 

13.  In  dividing  a  year's  profits  between  three  ])artners.  A, 
B,  and  C',  A  got  one-fourth  and  iM')<)  more,  Ji  got  one-third 
and  >^;)(i()  nu)rc,  and  C  got  one-lifth  and  -sGU  more.  What  was 
the  sum  divitled  ? 

14.  A  traveller  inquiring  the  distance  to  a  city,  was  told 
that  after  he  had  gone  one-third  the  distance  and  one-third 
the  remaining  distance,  he  would  still  have  'M\  miles  more  to 
go.     What  was  the  distaucc  of  the  city? 

15.  In  making  a  journey,  a  traveller  went  on  the  first  day 
one-lifth  of  the  distance  and  8  miles  more  ;  on  the  second  day 
he  went  one-fifth  the  distance  that  remained  and  15  miles 
more;  on  the  third  day  he  went  one-third  the  distance  that 
remained  and  \'l  miles  more  ;  on  the  fourth  he  went  '.^7^  miles 
and  finished  his  journey.  What  was  the  whole  distance 
travelled  ? 

16.  "When  two  i^artners  divided  their  profits,  A  had  twice 
as  much  as  B.  If  he  paid  B  'S;)()(),  he  would  only  have  half  as 
much  ag;iin  as  B  had.     W'hat  was  the  share  of  each  ? 

17.  At  noon  a  ship  of  war  sees  an  enemy's  merehnnt  vessel 
15  nules  away  sailing  at  the  rate  of  0  miles  an  hour.  How  fast 
must  the  shi])of  war  stiil  in  order  to  get  within  a  mile  of  the 
vessel  by  G  o'clock  ? 

18.  A  train  moves  away  from  a  station  at  the  rate  of  li 
miles  an  hour.  Half  an  hour  afterward  another  train  follows 
it,  running  \i\  miles  an  hour.  How  long  will  it  take  the  latter 
to  overtake  it  ? 

19.  AVhat  two  numbers  are  they  of  which  the  difference  is 
0,  and  the  difi'erence  of  their  S(iuares  ;)51  ? 

20.  A  man  bought  25  horses  for  $;35UO,  giving  $80  a  piece 


• 


ONE   UNKNOWN  QUANTITY 


103 


for  poor  liorso.s  and  ^i;JO  cacli  for  good  ones.     How  many  of 
each  kind  did  ho  huy  ? 

2  1.  A  man  is  T)  years  oldor  tlian  Ids  wit'o.  In  15  years  tlio 
sums  of  tlieir  aires  will  ho  three  linios  tlie  present  age  of  tho 
wil'u.     What  is  the  age  of  each '/ 

22.  ITow  f;ir  can  a  person  who  lia^  S  honrs  to  spare  ride  in 
a  coach  at  the  rate  of  0  ndlcs  an  honi'.  so  that  he  can  return  at 
the  rate  of  4  ndles  an  hour  ami  arrive  home  in  time? 

23.  A  working  alone  can  (h)  a  i)ieco  of  work  in  1.")  days, 
and  \\  alone  can  perform  it  in  1"^  days.  \\\  what  time  can  they 
perform  it  if  hoth  work  together  ? 

RIktiioi)  ok  Soi.ctiox,  In  one  day  A  rnn  do  -,'.t  of  tlio  whole  work 
and  B  ciin  do  i\^.     lltiucf,  both  togt'tlicr  can  do  (,'^,-!-  j'^,)  '>f  't. 

If  both  together  can  do  it  in  .r  days,  then  they  can  do     of  it  in  1  day. 


UA  +  1 

X       la      15 


Hence, 
is  tho  equation  to  bo  solved. 

24.  A  cistern  can  ho  filled  in  1'.'  minutes  1)y  two  ])i))es  which, 
run  into  it.  One  of  them  alone  will  till  it  in'  20  minutes.  In 
what  time  would  the  other  one  alone  iill  it? 

25.  A  cistern  can  he  em])lied  hy  three  pipes.  The  second 
pijie  runs  twice  as  much  as  the  first,  and  the  third  as  much  as 
the  lirst  and  second  together.  All  thive  together  can  emi)ty 
the  cistern  in  one  hour.  In  what  time  would  each  one  sepa- 
rately empty  it  ? 

26.  A  marketwoman  bought  apples  at  the  rfite  of  5  for  two 
cents,  and  sold  half  of  them  at  2  for  a  cent  and  the  other  half 
at  ;]  for  a  cent.  Her  profits  were  50  cents.  IIow  many  did 
she  buy  ? 

27.  A  grocer  having  50  pounds  of  tea  worth  90  cents  a 
pound,  mixed  with  it  so  much  tea  at  60  cents  a  pound  that 
the  combined  mixture  was  worth  TO  cents.  IIow  much  did 
he  add  ? 

28.  A  laborer  was  hired  voi  40  days,  on  the  condition  that 
every  day  he  worked  he  shouk'.  receive  $1.50,  but  slioidd  for- 
feit .50  cents  for  every  day  h^  was  idle.  At  the  end  of  the 
time  $52  were  due  him.     llow  many  days  was  he  idle  ? 

29.  A  father  left  an  estate  to  his  three  children,  on  the 
condition  that  the  eldest  should  be  paid  81200  and  the  second 
8800  for  services  they  had  rendered.  Tiie  remainder  was  to  bo 
C(pudly  divided  among  all  three.    Under  this  arrangement, 


<  I 

•  r  I, 


I 


it  i. 


104 


KQUATIONS   OF   Till':   FIRST  DKORFK. 


I'    k 


the  yoinipfcst  pot  oiic-fourlh  of  the  t'statc.     What  was   tho 
aiiiouiit  (lividi'd  ? 

30.  A  person  liavlnff  a  pum  of  nioiioy  to  divide  among 
three  people  ^^ave  the  first  (»ne-lliird  and  ^'l^)  inor(>,  tiie  second 
one-tliird  of  \vh:it  wns  hi't  and  ij^'JO  more,  and  the  third  one- 
third  of  what  was  then  hd'tand  ^^M)  more,  whicli  exiiautjLed  tho 
anionnt.     How  iniieii  had  they  to  divide  ? 

31.  One  slu'])herd  spent  $T'.M)  in  sheep,  and  anotliergot  tho 
same  nnmlter  of  sheep  for  84S(),  l)ayin^'  %i  a  pieee  less.  What 
])riee  did  each  l>a y  ^ 

32.  A  erew  wldeh  ran  pull  at  the  rate  of  0  miles  an  hour, 
finds  that  it  takes  twieo  as  \o\vr  to  ^o  \\\)  the  river  as  to  go 
down.     At  what  rate  does  the  river  flow  ? 

33.  A  person  who  possesses  !!<|t2()00  emjdoys  a  portion  of 
the  money  in  iaiildin*;  a  house.  01'  the  money  which  remains, 
lie  invests  one-third  at  four  per  eent.  and  the  other  two-tiiirds 
at  Hve  per  (!ent.,  and  ohtains  from  these  two  investments  un 
annual  ineome  of  %''V.)'l.     What  was  the  eost  of  the  house  ? 

34.  An  income  tax  is  levied  on  the  condition  that  the  first 
*r»0()  of  every  income  shall  he  untaxed,  the  next  ^;{()0()  shall 
be  taxed  at  two  percent,  and  all  incomes  in  excess  of  ^.'JOUO 
shali  he  taxed  three  per  eent.  on  the  excess.  A  i)erson  finds 
that  hy  a  uniform  tax  of  two  per  cent,  on  all  incomes  he  would 
save  $:^00.     What  was  his  income  ? 

35.  Atwluit  th.ie  between  W  and  1  o'clock  is  the  niinute- 
liand  5  minutes  ahead  of  the  hour  hand? 

36.  One  vase,  holdinjjj  a  jiallons,  is  fnll  of  water;  a  second, 
holding  b  gallons,  is  full  of  brandy,  l^'ind  tht'  cai)acity  of  a 
dii)per  such  that  whether  it  is  tilled  IVoju  the  first  vase  and  the 
water  removed  replaced  by  brandy,  or  filled  from  the  secotul 
vase  and  the  latter  then  fdled  with  water,  the  strength  of  the 
mixture  will  be  the  same. 

37.  Divide  a  number  m  into  four  such  parts  that  the  first 
part  increased  by  a,  the  second  diminished  by  rr,  the  third 
multiplied  by  a,  and  the  fourth  divided  by  ti  shall  all  he  e((ual. 

38.  Divide  o  dollars  among  five  brothers,  so  that  each  shall 
have  n  dollars  more  than  the  next  younger. 

39.  A  courier  starts  out  fnmi  his  station  riding  S  miles  an 
hour.  Four  hours  afterwards  he  is  followed  by  another  riding 
10  miles  an  hour.  Tfow  long  will  it  require  for  the  second  to 
overtake  the  first,  and  what  will  he  the  distance  travelled  ? 

If  X  be  the  nuiiibfr  of  hours  required,  tho  srrond  will  have  travelled 
X  hours  and  tin;  first  (j'4-4)  hours  whcu  they  meet.  At  this  timo  they 
must  have  travelled  equal  distances. 


OXE    UNKNOVrX  QUANTITY. 


105 


l*rohloiii  of  (Ik»  C'oiirh'rH, 

Let  us  jjonoruli/c  the  ])r('('<'(linnr  |)r<)])leni  tluis  : 

l.*M.  ./  rofirirr  sftn'fs  out  J'nnii  his  stnlimi  ritlin^  e 
inihs  <in,  liiKir ;  h  hours  Jnlrr,  he  is  ftilloii'rd  Inj  uiiothrr 
ridiiKj  ff  ini/rs  on  hour.  Ihur  I  on 'J  will  Ihr  In/frr  he  in, 
orrrfnkin^'  Ihr  jirsi ,  and  what  irill  be  the  distance  J'roni 
the  fioint  o/'  (li'p/trlnrc. 

Let  U8  put  /  for  tlio  timo  roqiiircd.  Tlu'ii  the  first  roiirii-r 
■will  liavo  tnivi'llcd  {l-\-lt)  hours,  juul  the  second  /  hourn. 
Since  the  lirst  travelled  r  miles  an  hour,  his  whole  distance  at 
the  end  of  t-\-li  hours  will  he  (l-\-/i)r.  In  the  same  way,  the 
distance  travelled  hv  the  other  will  he  (/I.  When  the  latter 
overtakes  the  former,  the  distances  will  he  e(|ual;  hence, 


at  =  c{l  +  h). 
Solving  this  equation  with  respect  to  /,  we  find 

t  =  -^'— 

a  —  c 


(1) 


(•i) 


Multiidying  l)y  a  gives  us  the  wliole  distance  travelled, 
which  is 

Distance  =         -• 

a  —  c 

This  equation  solves  every  prohlem  of  this  kind  hy  suhsti- 
tuting  lor  a,  c,  and  h  their  values  in  numhcrs  su])posed  in  the 
prohlem.  For  example,  in  Prohlem  39,  we  supposed  a=  10, 
c  =  8,  h  :=  4.  Substituting  these  values  in  e(puition  (:i),  we 
find 

t  =  in, 

which  is  the  nnnihcr  of  hours  required. 

To  illustrate  the  generality  of  an  algehraic  prohlem,  we 
shall  now  inquire  what  values  t  shall  have  when  we  make  dif- 
ferent suppositions  respecting  a,  c,  and  h. 

(1.)  Lot  us  suppose  ((  =  r,  or  a  —  c—=0,  that  is,  the  rates 
of  travelling  equal.    Then  equation  (5i)  will  become 

nh 
'-  0' 


ill 


I 


106 


FQUATIO 


OF   THE  FIRST  DEGREE. 


f 


an  cxprcssioh  for  infinity  (§  112,  G),  showing  tliat  tho  oiio  oonrier 
would  never  overtake  the  other.     This  is  phiiu  enough.     liiit, 

(2.)  Let  us  supjiosc  that  the  second  courier  does  not  ride 
so  fast  as  the  first,  that  is,  a  less  than  c,  and  a  —  c  negative. 

Tlien  the  fj-action will  not  be  infinite,  but  will  Ijc  nejra- 

a  —  c  ° 

five,  because  it  has  a  positive  numerator  and  a  negative  denom- 
inator. It  is  plain  that  the  second  courier  would  never  o\  ertakc 
tlie  first  in  this  case  eitlier,  because  the  latter  would  gain  on 
him  all  the  time  ;  yet  the  fraction  is  not  infinite. 

AVhat  does  this  mean  ? 

It  means  that  the  })ro])lem  solved  by  Algebra  is  more  gcn- 
ci'al,  that  is,  involves  more  particular  proljlems  than  were 
im])lied  in  tho  statement.  If  we  count  the  hours  ((fler  the 
second  courier  set  out  as  positive,  then  a  negative  time  will 
mean  so  many  I, ours  before  he  set  out,  and  this  Avill  ])ring  out 
a  time  when,  according  to  our  idea  of  the  problem,  the  horses 
were  still  in  the  stable. 

The  exi)lanati(m  of  the  difficulty  is  this.     Sujipose  S  to  be 

the  point  from  which  the  couriers  started,  and  AB  the  road 

along  which  they  travelled  from 

AS 

S  toward  r>.     Suppose  also  that         .^-i.^— »..»...^^_ 

the  first   courier    started    out 

fi'om  S  at  8  o'clock  and  the  second  at  13  o'clock.  By  the  rule 
of  positive  and  negative' quantities,  distances  towards  A  arc 
negative.  Now,  because  algebraic  ([uantities  do  not  commence 
at  0,  but  extend  in  l)oth  the  negative  and  positive  directions, 
the  algebraic  problem  does  not  suppose  the  couriers  to  have 
really  C(mnnenced  their  journey  at  S,  but  to  have  come  from 
the  direction  of  A,  so  (hat  the  first  one  passes  S,  without  stop- 
l»ing,  at  8  o'clock,  aiul  the  second  at  12.  It  i<  })lain  that  if  the 
first  courier  is  travelling  the  faster,  he  must  have  passed  the 
other  before  reaching  S,  that  is.  the  time  and  distance  arc 
both  negative,  just  as  the  problem  gives  them. 

'^rhe  general  i)rin('l[)le  here  involveil  may  he  ex})resscd  thus: 
In  Ahji'bra,  roads  and  journeys,  Ui-e  time,  have  no  hcfjln- 
ni)H)  and  no  end. 


B 


V 


t 


•^^^T 


ONE    UNKNOWN  QUANTITY. 


107 


■ 


t 


(3.)  Let  us  suppose  that  tlie  couriers  start  out  at  the  same 
time  and  ride  Avitli  the  same  speed.  Tlieii  h  and  a  —  c  aro 
both  zero,  and  the  expression  for  t  assumes  the  form, 

0 


t  = 


0 


This  is  an  expression  whieh  may  have  one  vaUic  as  well  as 
anotlier,  and  is  therefore  indeterminate.     The  result  is  correct,  i 
because  the  couriers  arc  always  together,  so  that  all  values  of 
t  arc  equally  correct. 

The  equation  (1)  can  be  used  to  solve  the  problem  in  other 
forms.  In  this  ecpiation  are  four  quantities,  a,  c,  h,  and  I,  and 
when  any  three  of  those  aro  given,  the  fourth  can  be  found. 
There  are  therefore  four  problems,  all  of  which  can  be  solved 
from  this  equation. 

First  Problem,  that  already  given,  in  which  the  time 
required  for  one  courier  to  overtake  the  other  is  the  unknown 
quantity. 

Secoxd  Problem.  A  courier  sets  out  from  a  station, 
riding  c  miles  an  hour.  After  It  hours  another  follows 
him  from  the  same  station,  intending  to  overtake  Jiinv 
in  t  hours.    How  fast  must  he  Hde? 

The  problem  can  be  put  into  the  form  of  an  equation  in 
the  same  way  as  before,  and  we  shall  have  tlie  equation  (1), 
only  a  will  now  be  the  unknown  quantity.  If  we  use  the 
numbers  of  Prob.  39  instead  of  the  letters,  we  shall  have,  in- 
stead of  equation  (1),  the  following  : 

1G«  =  8  (IG  +  4)  =  8-20  =  IGO, 

whence  «  =  10. 

If  we  use  letters,  we  find  from  (1), 

c{t-\-h) 


a  = 


t 


and  the  problem  is  solved  in  either  case. 

Third  Problem.     Tlie  second  courier  can  ride  just  a 
miles  an  hour,  and  the  first  courier  starts  out  h  hours 


!  I 


I 


108 


EQUATIONS  OF  TUE  FIRST  DEGREE. 


*» 


he  fore  Jdni.     IIow  fast  must  the  latter  ride  in  order  that 
tlie  other  may  take  t  hours  to  overtake  him? 

Hcc  c,  the  rate  of  tlie  first  courier,  is  the  uuknown  quan- 
tity, and  hy  solving  equation  (1),  we  find 

at 

Fourth  Problem.  The  swiftest  of  tivo  couriers  can 
ride  a  luiles  an  hour,  and  the  slower  c  miles  an  hour. 
How  long  a  start  must  the  latter  have  in  order  that  tlie 
other  may  require  t  hours  to  overtake  him? 

Here,  in  equation  (1),  h  is  the  unknown  quantity.  By 
solving  the  equation  with  respect  to  h,  we  find, 

,        at  —  ct 
which  solves  the  problem. 

PROBLEMS    OF    CIRCULAR    MOTION. 

40.  Two  men  start  from  the  same  point  to  rnn  repeatedly 
round  a  circle  one  mile  in  circumference.  If  A  runs  7  miles 
an  hour  and  B  5,  it  is  required  to  know : 

1.  At  what  intervals  of  time  will  A  pass  B  ? 

2.  At  how  many  different  points  on  the  circle  Avill  they  be 

together  ? 

We  reason  thus  :  Bince  A  runs  2  miles  an  hour  faster  than  B,  he  cets 
away  from  him  at  tlie  rate  of  3  miles  an  hour.  When  he  overiakes  him, 
he  will  have  gained  up  m  him  one  circumference,  that  i.s,  1  mile.  This 
will  require  ;}0  minutes,  whicli  is  therefore  the  refjuired  interval.  In 
this  interval  A  will  have  gone  round  iJ^  and  B  2i  times,  so  that  they  will 
be  together  at  the  point  opposite  that  where  they  were  together  30 
minutes  previous.  Hence,  tliey  are  together  at  two  opposite  points  of 
the  circle. 

41.  AVhat  would  be  the  answer  to  the  preceding  ques- 
tion if  A  should  run  8  miles  an  hour,  and  1>  5? 

42.  Two  race-horses  run  round  and  round  a  course,  the 
one  makiuij:  tlie  circuit  in  30,  the  other  in  35  seconds.     If 
they   start    out 
toixethcr  airain  ? 

Note.    In  x  seconds  one  will  make  ..  -  circuit  and  the  other  _=. 

43.  If  one  ]tlanct  revolves  round  the  sun  in  T  and  the 
other  in  T'  years,  what  will  be  the  interval  between  their 
conjunctions? 


together, 


how    long    before    they  will    be 


*i 


TWO    U:S KNOWN  QUANTITIES. 


109 


CHAPTER    III. 

EQUATIONS   OF   THE    FIRST    DEGREE    WITH    SEVERAL 
UNKNOWN    QUANTITIES. 


Case  I.  Equations  ivith  Two  TJnknowii  Quan- 
tities. 

133.  Def.  An  equation  of  tlie  first  degree  with  two 
unknown  quantities  is  one  which  admits  of  being  re- 
duced to  the  form 

ax  -^b?/  =  c, 

in  whicli  x  and  ?/  are  tlie  unknown  quantities  and  a,  ?;, 
and  c  represent  any  numbers  or  cxlg(d)raic  equations 
which  do  not  contain  either  of  the  unknown  quantities. 

Def.  A  set  of  several  equations  containing  tlie  same 
unknown  quantities  is  called  a  System  of  Simulta- 
neous Equations. 

Solution  of  a  Pair  of  Siimiltaneoiis  Equations 
containing^  Two  Unknown  Quantities. 

l.*>3.  To  solve  two  or  more  simultaneous  equations, 
it  is  necessary  to  combine  them  in  such  a  way  as  to 
form  an  equation  containing  only  one  unknown  quan- 
tity. 

134.  Def.  Tlie  process  of  combining  equations  so 
that  one  or  more  of  the  unknown  quantities  shall  dis- 
ap]x\'ir  is  cjiILmI  Elimination. 

The  term  "elimination"  is  used  because  tlic  unknown 
quantities  which  disapi)ear  arc  rliniiua/cd. 

There  arc  tliree  methods  of  eliminating  an  unknown  (|nan- 
tity  from  two  ^iiniullaneous  equations. 


110 


EQUATIONS   OF   THE  FIRST  DEGREE. 


Eliiiiiiiiitioii  by  Comparison. 

135.  Rule.  Solve  each  of  the  cqiuitions  irith  rcftpcct 
to  one  of  the  luihiiouni  (iiudititles  (nul  piittJte  two  values 
of  the  iLiikiioivii  qaaiititij  thus  ubtaliicd  equal  to  each 
other. 

This  ivill  i>ij'e  an  eqaatioji  iritJv  only  one  ujiA'jiojvji 
quantltij,  of  ivhleh  the  value  can  he  found  from  tJte 
equ,ation. 

Tlie  valm  of  the  other  unknown  quantity  is  then 
found  hy  substitution. 


Example.     Let  the  cqiuitions  be 

ax  -\-  hj  =.  c,  \ 

ax  +  h'y  ■=  c.  j 

From  the  first  equation  wc  obtain, 

C  —  1)11 

X  = -' 


a 


From  the  second  we  obtain, 


~~       a' 
Putting  these  tAvo  values  equal,  we  have 

c  —  hf  c'  —  l)'y 


(1) 


(2) 


(3) 


a 


a 


Reducing  and  solving  this  equation  as  in  Chapter  II,  we 
find, 


y 


ac  —  ae 
ab'^^aV 


which  is  the  required  value  of  y.    Substituting  this  value  of  y 
in  eitlicr  of  the  equations  (1),  (2),  or  (3),  and  solving,  we  shall 

find 

I'c  -  hr' 

ah'  —  ab 

If  the  work  is  correct,  the  result  will  be  the  same  in  which- 
ever of  the  equations  we  make  the  substitution. 


X  = 


¥l 


¥^ 


TWO    UNKNO}yN  QUANTITIES.  Ill 

Numerical  Example.     Let  tlic  equations  bo 

3.1-  —  'Zy  =  20.  f  ^*' 

From  the  first  cqiiiition  we  find 

X  =  -^S  —  Iff 

and  from  the  second  x  =  - — - — '- , 

o 

Of)     I     Ow 

from  which  wc  have  28  —  ?/  =  -— ., — -, 

ij  =  11. 
Substituting  tliis  vaUie  in  tlio  first  equation  in  x,  it  becomes 

a:  =  28  -11  =  17. 

If  Ave  substitute  it  in  the  second,  it  becomes 

20  +  22        51 
X  = ., =  ^  =  17, 

tlie  same  vahie,  thus  proving  the  correctness  of  the  Avorl^:. 


( 


EliiiiiiiJitioii  by  Substitution. 

13(5.  Rule.  Find  the  value  of  one  of  the  unhnouii 
qunntitics  in  terms  of  the  other  from  cither  cijiudioii, 
and  snbstitnte  it  in  the  other  eqaation.  Hie  latter  will 
have  hut  one  unknown  quantitij. 

Example.    Taking  tlie  same  jquations  as  before, 

ax  -\-  hij  =  <7, 
a'x  +  V  y  =  c, 

the  first  equation  gives        x  ■=. 


a 


Substituting  this  value  instead  of  x  in  the  second  equation, 

it  becomes 

a'c  —  a'hy 


a 


+  ////   rr   C'. 


Solving  this  e(iuatiou  with  respect  to  y,  we  get  the  samo 
result  as  before. 


112 


EQUATIONS   OF  THE  FIRST  DEGREE. 


>  ! 


Numerical  Example.    To  solve  in  this  way  the  lust  nu- 
mcricjil  example,  wc  have  from  the  tirst  equation  (-1), 

X  =  'l^  —  ij. 

Substituting  this  value  in  the  second  equation,  it  becomes 

84  —  3//  -  2y  =  29, 

from  which  we  obtain  as  before, 

84  —  -30 


y  = 


'-  =  11. 


This  method  may  be  applied  to  any  pair  of  equations  in 
four  ways  : 

1.  Find  X  from  the  first  equation  and  substitute  its  value 
in  the  second. 

2.  Find  X  from  the  second  equation   and   substitute   its 
value  in  the  first. 

3.  Find  y  from  the  first  equation  and  substitute  its  value 
in  the  second. 

4.  Find  y  from  the  second  equation   and   substitute   its 
yalue  in  the  first. 


ti 


*» 


Elim illation  by  Addition  or  Subtraction. 

13*7.  Rule.  Midtipln  each  equation  hij  such  a  factor 
that  tJi'6  cocjficlciits  of  one  of  the  unkiwwn  quantities 
shall  hccoDie  uiimerically  equal  in  the  two  equations. 

TJien,  by  adding  or  subtracting  the  equations,  ire 
shall  have  an  equation  luitli  but  one  unknown  quantity. 

Rem.  AVc  may  always  take  for  the  factor  of  each  erjuation 
the  coefficient  of  the  unknown  quantity  to  Ix:  eliminated  in  the 
other  equation. 

Example.    Let  us  take  once  more  the  general  equation 

ax  -\-hy  =  c, 
a'x  -\-  by  =  c. 

Multiplying  the  first  equation  by  a  ,  it  becomes 

aax  +  ahy  =  a'c. 
Multiplying  the  second  by  a,  it  becomes 

aa!x  -\-  ah'y  =  ac. 


i 


TWO    UNKNOWN  QUANTITIES. 


113 


The  unknown  quantity  x  has  the  same  cntfiicicnt  in  the 
last  two  equations.     Subtracting  them,  from  each  other,  we 

obuia 

{a'b  —  ah')  y  =z  a'c  —  ac\ 


y  = 


a  0 


ac 


a'b  —  ati 


Rem.     "We  shall  always  obtain  the  same  result,  whichever 
the  above  three  methods  we  use.     But  as 
last  method  is  the  most  simple  and  elegant. 


of  the  above  three  methods  we  use.     But  as  a  general  rule  the 


Problem  of  the  Sum  and  Difference. 

The  following  simple  problem  is  of  such  wide  application 
that  it  should  be  well  understood. 

138.  Problem.     T]i6  sum  and  difference  of  two  num- 
hers  bein'Ji  given,  to  find  the  nunihci's. 

Let  the  numbers  be  x  and  y. 

Let  .V  be  their  sum  and  d  their  ditTerence. 

Then,  by  the  conditions  of  the  problem, 

x-\-y  =  s, 
X  —  y  ■=  d. 
Adding  the  two  equations,  we  have 

2a;  =  .s  -|-  d. 
Subtracting  the  second  from  the  first, 

'ly  =z  s  —  d. 
Dividing  these  equations  by  2, 


X 

= 

s  +  d 

2 

z=. 

s 
2 

-t 

y 

= 

s  —  d 
2 

= 

2 

d 
2 

We  therefore  conclude : 

JJie  greater  mtnihcr  is  found  hy  adding  half  the  dif- 
ference to  half  the  sum. 

Hie  lesser  iiujuher  is  found  hy  subtracting  half  the 
difference  from  half  the  sum. 
8 


.Ji 


114 


EQUATIONS   OF   TUB  FIRST  DEGREE. 


i 


B 


C 

I 


I 
P 


This  result  can  be  illiistrjitcd  geometrically.  Let  AB  and 
BC  be  two  liiK's  ])laced  end  to  end,  so  that  AC  is  their  .sum. 
To  find  tlu'ir  (lilTcienee,  we 

cut  oil"   from  AB  a  lengtli         , 

AC  —  BC  ;  then  C'B  is  the 
dillerenee  of  the  two  lines. 

If  P  is  half  way  between  C  and  B,  it  is  tie  middle  point 
of  the  whole  line,  so  tliat 

AP  =  PC  =  I^AC  =  I  sum  of  lines. 
C'P  =  PB  =  iC'B  =  i  dill'erenco  of  lines. 

If  to  the  half  sum  AP  we  add  tlie  half  difference  PB,  "we 
have  AB,  the  greater  line. 

If  from  tlie  half  sum  AP  we  take  the  half  diilereucc  C  P, 
■WG  have  left  AC,  the  lesser  line. 


th 


'^,^^ 


wh( 


'» 


EXERCISES. 

Solve  the  following  equations : 


I. 

2. 

3- 
4. 
5- 
6; 


8. 


3.K 


3.r 


6"^  7 


4^5 


9.y  =  33,     2.?;  —  3;/  r=  18. 
by  —  13,     %€  +  7//  =z  81. 

Gx  -\-  i')i/  =  Z». 

2x  —  3y  =  w. 

ax  —  hi/  =  q. 

^    y  _  o 
^  ,y  _ 

8^2 


7y  +  Qy  =:  a, 
2x  +  3//  =  7)1, 
ax  -\-Jjy  ~  p, 

=  20, 


=  18, 


20. 


y 


2+3    =^' 


X 

2 


y 


-     =h. 


9-      <  (•'•  +  ?/)  +  3  {x  -  y)  ^'^  102, 
7{x  +  y)-^{x-y)  =GG. 

Note.  Solvo  tins  equation  first  as  if  .r  +  ?/  and  x—y  were  sinfrle  syni- 
bois.  of  which  the  values  are  to  be  found.  Then  find  x  and  y  by  §  138 
l)rec'ediiig. 


10. 


X 


+  //  +  ('^  -  y)  =  14'  ^  +  2/  —  (-^  -  2/)  =  10- 


I 


6ec( 


wh 


II. 


x 


+^= 


X 


2/ -12 


TWO    UNKNOWN  QUANTITIES. 


115 


id 


u. 


it 


'G 


Note.     Equations  in  this  form  can  bo  best  solved  as  if  -  and      wero 


the  uuiinowu  (luantitius.     See  next  exercise 

3_2 

X       y 


12. 


5 


10'     x'^y-'^' 


Solution.    If  we  multiply  the  first  equation  by  4,  and  the  second  by 
3,  we  have 

12_  8    _  44  _  22 

5' 


ss 


y 


10 


12       15        .        45 

r=  'J  =  -v" 

x         y  0 

Subtracting  the  first  from  the  second,  we  have 

23  _  23 

y  ~  ^' 

whence, 

y  =  5. 

Again,  to  eliminate     ,  we  multiply  the  first  equation  by  5  and  the 
second  by  2  and  add.     Thus, 

15  _  10  _  n 

X     7  -  T' 

8       10        .        12 


whence, 

13. 
14. 

15- 


17. 


23        23 

»  ~"  2  ' 

X  =  2. 

X       y 

723              1 
~  12'     X      y~       12 

'  +  '- 

X      y 

5       2       15 
"~  12'     X      y  ~  24' 

5       3 

X      y 

13       11 

""        G'     «       2/  ~~  30* 

5 

3                 1          3 

1 

X  -\-l 

y  —  l~       G'    .-r  +  1 

y-l 

2 

3      _   7           2 

3 

30 


a:  + 


?/  — 3       12'    a; +  2      y  — 3 


12 


110 


EQUATIONS  OF  THE  FIU6T  JJEGIiEE. 


1 8.        4-      =  c, 

X      y        '     X      y 


19. 


20. 


a      b        J 
—     =  a. 


X 


X  —  y         '       X  -\-  a 

a^b^  a-b~    "'      ^ab    "  ^' 


Case  IT.  !Ef/Kftffon.9  of  the  First  jycffree  with 
Three  or  More  Unknotvn  Quantities, 

139.  Wlion  the  viiluos  of  scvcnil  unknown  quantities  arc 
to  be  found,  it  is  necc.'^sjiry  to  luivc  as  many  equations  as  un- 
known ([uantities. 

If  there  are  more  unknown  quantities  than  equations,  it 
will  be  impossible  to  determine  the  values  of  all  of  them  from 
the  equations.  All  that  can  be  done  is  to  determine  the  value 
of  some  in  terms  of  the  others. 

If  the  number  of  e([uations  exceeds  that  of  unknown  quan- 
tities, the  excess  of  equations  will  be  superlliious.  If  there 
are  n  unknown  quantities,  their  values  can  be  found  from  any 
n  of  the  equations.  If  any  selection  of  n  equations  we  choose 
to  make  gives  the  same  values  of  the  unknown  quantities,  the 
equations,  though  superfluous,  will  be  consistent.  If  different 
values  tire  obtained,  it  will  be  impossible  to  satisfy  them  all. 


'I 


Eliiiiiiiatioii. 

140.  "When  the  number  of  unknown  quantities  exceeds 
two,  the  most  convenient  method  of  elimination  is  generally 
that  by  addition  or  subtraction.  The  unknown  quantities  arc 
to  be  eliminated  one  at  a  time  by  the  following  method  : 

I.  Select  an  milaioiun  quantity  to  he  first  cliniinafed. 
It  is  best  to  begin  irith  the  qnantitij  wliicli  appears  in 
the  fewest  equations  or  has  the  simplest  coefficients. 

II.  Select  one  of  the  equations  containing  this  un- 
known quantity  as  an  eliniinatii/g  equation. 

III.  Eliminate  the  quantity  bctu'ccn  this  equation 
and  each  of  the  others  in  succession. 


f^ 


TUREE  on   MORE    UNKNOWN  QUANTITIEii.       117 


^- 


Wc  shiiU  then  liiive  a  socoiul  system  of  equations  loss  by 
one  ill  nunihtT  than  the  ()ri<,'iiiiil  system  and  containing'  a  num- 
ber of  unknown  quantities  one  K'ss. 

IV.  liCjM'nt  the  /jfoccssi  uu,  the  new  fujsteni  of  equation  a, 
(Hid  continue  tlie  rejxtition  until  oniijoncequdtion  wilk 
one  unknown  qnuntitij  Is  left. 

V.  IfariinJ  found  the  vulue  of  this  hist  unknown 
fjuftntiti/,  the  ludues  of  the  others  eunl/e  found  Ijij  suC' 
cessive  suhstitittion  in  one  equcttiun  of  each  system. 

Example.     Solve  the  equations 

(1)  4:c  — 3^—    z+    u—    7  =  0, 

(3)  X—    v/  +  22  +  'in  —  10  =  0, 

(3)  ^x  +  2f/  —    z  —  'ln—    2  =  0, 

(4)  a;  4-  2y  +    2;  +    u  —  19  =  0. 

We  shall  seloct  .t  as  tlu;  first  quantity  to  be  eliminate' I,  and  take  tlie 
last  ('(jiiatioii  as  the  elimiiiiitiiii^^  one.  Wo  first  multiply  iliis  ecjuation  by 
three  such  factors  that  the  cocincieut  of  X  shall  become  ('(jiial  to  the  co- 
eilicient  of  x  in  each  of  the  other  e(|uations.  These  factors  are  4,  1,  and  2. 
We  write  the  products  under  each  of  the  other  ecjuations,  thus  : 


(«) 


7  =  0, 


4./;  —  3//  —    z  -\-    u  - 

4x  +  K'/  +  -12  +  4w  —  70  =  0. 


X  —    ?/  +  2z  -\-  2u 
X  4-  2//  -{-    z  \-    V 

—  10  =  0, 

—  10  =  0. 

^x  +  2}/  —    z  —  2u. 

2x  +  4//  -^  2z  +  2u 

-2  =  0, 
—  38  =  0. 

Eq.  (1), 

(4)  X  4, 

E(i.  (2), 
(4)  X  1, 

Eq.  (3), 
(4)  X  2, 


By  subtracting  the  one  of  each  i)air  from  tlie  otlier,  we  obtain  the 
equations, 

11^  +  52  +  ^u  —  09  =  0,  \ 
3//  —    z—    u—    9  =  0,  V  (J) 

2^  +  3;^  +  42t  —  30  =  0.  ) 

The  unknown  quantity  x  is  here  eliminated,  and  we  have  three  equa- 
tions with  only  three  unknown  (luantities.  Now  eliminatin<r  ,v  by  means 
of  the  last  equation,  in  the  same  way,  and  clearing  of  fractions,  we  find 
the  two  equations, 


23^;  +  38?^  —  258  =  0, 


Uz  +  Uu  —    90 


0. 


(P) 


(a 


m\ 


%^ 


!i 


118 


EqUATIONS   OF   THE   FlIiST   DEdltKK. 


Till'  prolilcin  Ih  now  rcdiuvMl  t<i  two  ('<|iiatinnM  with  two  unknown 
quiiMtilitH,  whicU  w«)   liavo  ulrcuily  bIiowu  how  to  Molvr.      We  (iml  liy 

ttolvillj^r  tllt'lll, 

« =  —  a, 

11  —  8. 

Wo  next  find  the  value  of  y  liy  Hulistitiitin^'  these  valiicM  of  z  and  it, 
in  cither  of  tlic  c'liinitioiiH  {h).     The  lii'Ht  of  them  thuH  becuna'tf ; 


11^  _  10  +  2-4  — GO  =  0, 


from  whieli  wo  find, 


y 


5. 


We  now  Huhstitufe  the  values  of  y,  ?,  and  u  in  pith(>r  of  equations  (a), 
riio  second  of  the  latter  heconjes 

a:  _  5  _  -1  4-  IG  —  10  =  0, 

and  the  fourth  becomes, 

:,..  4-  10  —  J2  -t-  8  —  m  =  0, 

either  of  which  gives 

X  —  3. 

We  can  now  yirnvo  the  results  hy  suhstitutinpf  the  values  of  .t,  //,  ?, 
and  H  in  all  ftnir  of  ciiuatiuns  {a),  and  seeing  whetlier  they  are  all  satislicd. 


'I 


EXERCISES. 

1.  One  of  tlic  ))Ost  exorcises  for  llie  sliuleiit  Avill  ])e  tliat  of 
resolviiiof  the  i)revioiis  e(|iiiitions  {a)  hy  takiii;,^  the  last  c'(|ua- 
iioii  as  the  eliiniiialni.i,'  one,  {iiul  perform iiiiif  (he  eniiiiiiatioii 
ill  (litTereiit  orders;  that  is,  he^nii  hy  eliiiiiiialinu:  u,  then 
rejx'at  I  lie  whole  ])ro(ess  heginniiig  with  z,  etc.  The  liiiul 
results  will  alwavs  he  the  same. 

2.  Find  the  values  of  x^,  a-„,  :i\,  and  x^,  from  the  ccfua- 
tions, 

x^  +  ^\  +  >f  3  +  ^'4  =  ^^-i' 
^1  +  ^\  —  ^%  —  ^\  =  34, 

ft'j    Xn    H~   ^3    X ^     ^=        O, 

X\   —  X n  X-^   -\-  X /^    =^      4. 

This  oxamplo  requires  no  nuiltii)licatiou,  but  only  addition  and  sub- 
traction of  the  dill'ercnt  cMiuatious. 

3.  'Ix  -f  5?/  +  3^  =  13, 

'Ix  +  ^?,y  —    2;  =  12, 
hx  +.by  —  'Zz  —  21). 


1 


PliOULEMS. 


Hi) 


'.U'  -f-  //  —  \u  —    0, 
.,.  +  7^  _  «;//  =  ;j;{, 

:)Z  —  )lx  —  8y  f-  :.'/«  =  15. 


X  +  1/  +  z  =  (I, 
Z  -\-  U  ■{•  X  :=  Cy 

u  -\-  X  +  1/  =  d* 


6.        —      =  ;/?, 


—     =  «, 


X 

y 

1 

1 

.'/ 

2; 

1 

1 

« 

+ 

X 

=  1>- 


\i\ 


I  of 
|U;i- 
ioii 
lien 
iiKil 


PROBLEMS    FOR    SOLUTION. 

1.  A  man  liad  ii  saddlo  \V(irth  81")  uiul  twD  liorscs.  Tf  llio 
Siuldlc  he  pill  on  horse  A  lu'  will  he  doiildi-  ihc  value  of  H,  itiit 
if  it  l)e  piit  oil  !>  his  value  will  bu  ctiuul  to  thai  oi' A,  What 
is  tliu  value  of  eaeh  horse  ? 

2.  What  nuniher  of  two  di^i^its  is  equal  to  7  times  th^  sum 
of  its  digits,  and  to  'Zi  times  the  dill'ereiice  of  its  digits!' 

Lot  X  1)0  tlio  firnt  (li^nt,  or  the  iiiimbor  of  tens,  niid  //  tlio  unltH.  Then 
the  nuniluT  itself  will  be  10.r  +  //.  Seven  tinu-s  the  sum  of  the  dibits  iin^ 
7.r  +  7//,  1111(1  'i\  tiiiii'S  the  (liflcrence  are  21/— ;21//.  Uiiitiii^f  iiiio  solviiijj 
the  equations,  wo  find  j;  =  0,  y  =  ;j ;  the  muuljor  is  therefore  01'. 

3.  A  inimher  of  two  digits  is  equal  to  0  times  the  sum  of 
its  digits,  ami  if  U  be  subtracted  from  the  number  the  digits 
are  reversed.     What  is  the  number? 

4.  Find  a  numl)erof  two  digits  sueh  that  it  shall  be  equal 
to  G  times  the  sum  of  its  digits  inereasi'd  by  1,  whili'  if  IS  be 
subtracted  from  the  number  the  digits  will  f)e  reversed. 

5.  Find  a  number  Avhich  is  greater  by  2  than  5  times  the 
sum  of  its  digits,  and  if  9  be  added  to  it  the  digits  will  be 
reversed. 

6.  What  number  is  that  whicli  is  equal  to  0  limes  the  sum 
of  its  digits  and  is  -4  greater  than  11  times  their  ditlerenceV 

7.  What  fraction  is  that  which  becomes  e(|ual  to  f  when 
the  numerator  is  increased  by  ^,  and  e(|ual  to  ^  when  the  de- 
nominator is  increased  by  4. 

8.  Two  drovers  A  and  B  went  to  market  witli  cattle.  A 
sold  50  and  then  had  left  half  as  many  as  H,  who  had  sold 
none.  Then  15  sold  54  and  had  remaining  half  as  many  as  A. 
Ilow  many  did  each  have  ?  • 


nJ! 


120 


EQUATIONS  OF  TlU']  FIRST  DFAUIEE. 


'f 


9.  A  Itoy  hou^Mil  I'i  :i|)|>l<'s  l'(U-  a  (li)lljir,  jj^iviii*?  .'J  rrnis  onch 
for  (lie  <j^()()(l  (»iirs  Mild  'J  ('('nls  I'licli  lor  (lie  pcior  ones.  JIow 
iiiaiiy  of  each  kind  did  lie  l)iiy? 

10.  I''ind  a  IVaclion  wliicli  becomes  e(|iial  to  \  when  i(>? 
(leiioniinalor  is  increased  Ity  i;j,  and  to  5  when  4  is  snbli'acU'd 
fnnn  lis  nnnieralor. 

1 1.  I"'iiid  a  IVaclion  which  will  lieeonie  0(|nal  lo  3  hyaddin<j; 
"i  lo  ils  nuinenilor,  or  hy  adding  l.o  its  deiioniinalor  \\,  will  he- 
come  J^. 

12.  A  Jiuclvster  hon«i;lit  a  certain  numher  of  «hickens  at; 
',Vl  (HMits  (>a,ch  and  of  turkeys  at.  75  cents  each,  payin^^  !i?l  I  for 
till'  whole.  lie  sold  the  chickens  at  IS  cents  each,  and  the 
tnrkeysat  ^\  t>acli,  reali/ini,'  ^'li)  for  the  whole.  Jlow  many 
c'hit'kens  and  how  many  turkeys  had  he  ? 

13.  An  apjiU'woman  l)ou<;ht  a  lot  of  a|>|)les  at  1  cent  each, 
and  a  lot  of  pears  at  2  cents  each,  payin^j^  J^l.TO  for  the  whole. 
1 1  of  the  apjiK's  and  7  of  (he  pears  were  had,  hut  she  sold  the 
<:ood  a|)pli>s  at  '^  cents  i>;icli  and  the  <;ood  pears  at  ;{  centseach, 
realiziui^  i^'J.CiO.      Now  many  of  each  fruit  did  slu'  buy? 

14.  Wlu'u  Mr.  Smith  was  marrit'd  he  was  \  olih>rtlian  his 
Avil'i' ;  twi'lve  years  al'terward  ho  was  \  older.  What  were  their 
ages  when  married  ? 

15.  A  and  W  toiivlher  can  do  a  jtiece  of  work  in  (>  days,  hut 
A  workiuiX  alone  can  ^\o  it  !>  days  sooner  than  B  workin<5 
aloni'.     In  what  time  could  I'ach  of  them  do  it  sinu^ly  ? 

i(),  A  husband  beini^  asked  the  a<j:e  of  himself  and  wife, 
replied:  **lf  you  divide  mv  a_«::e  (1  years  hence  by  her  ai,a^ 
(i  yeai's  ago,  the  ([uotient  will  he  'i.  Hut  if  you  divide  her  ago 
Iv'  years  hence  bv  nunc  'IV  years  ago.  the  (luotient  will  be  5. 

^"•.  The  sum  of  two  aues  is  it  times  their  ditfercnce,  hut 
seven  years  ago  it  was  only  sovoii  times  their  ditrerence.  What 
are  the  ages  now  ? 

iS.  Two  trains  set  out  at  the  same  moment,  the  (me  to  go 
from  Boston  to  Springlield.  the  other  from  Sj>ringlield  to  Bos- 
ton. The  distance  between  the  two  cities  is  !>S  miles.  They 
meet  each  other  at  theend  of  1  hr.  "24  min..  aiul  the  train  from 
lit>stou  travels  as  far  in  4  hrs.  as  theotlier  in  3.  What  was  llio 
siJoeil  of  eai'h  train  ? 

K),  A  grocer  bought  50  lbs.  of  tea  and  100  lbs.  of  cotTeo  for 
$•10.  He  sold  the  tea  at  an  advanci>  of  }  on  his  price,  and  the 
cotTi'c  at  an  advance  of  J^,  realizing  *T 7  from  both.  At  what 
])rice  per  pound  did  ho  buy  and  sell  each  article  ? 

NoTK.  If  X  ami  y  are  the  prices  at  which  he  bought,  theu  \x  aud  ly 
are  the  prices  at  which  lie  sold. 


INCn NSISTF.NT    EQ  UA  TfONS. 


121 


IH'- 


20.  Im)!' />  (lolliirs  I  ran  ptirclmsc  cillicr  ^r  poimd.s  of  lea  and 
h  piinnds  ol'  colTcc,  or  ni  pounds  of  lea  ujid  //  })(<und.s  <)!"  (;oilc((. 
\Vlia(  is  (lie  price  per  pound  orcacli  ? 

2r.  A  ;^n»ldsrnilli  luid  (wo  in;:;o(s,  The  (irsi  is  composed  (d" 
c(pial  parlsoT^old  and  silver,  while!  the  second  contains. ")  parts 
)d' ^oid  t>o  1  ol'silver.  lie  watits  to  tak(!  from  tJieni  a  walch- 
cas(!  hiivinii^    1   ounces  of  (-old  and    I  oiinco  of  silver.     Jlow 


^t  he  tuko  fr 


om  eacli  in^^ot, 


mucii  niusl  lie  iuko 

22.  A  I)ankor  lias  two  kinds  of  (;oin,  sucli  tliah  a  ])ieres  of 
tlie  lirsi  kind  or  b  j)ieces  of  the  second  will  luake  a  dollar,  if 
he  wants  lo  select  c  i)ieces  which  shall  be  woilii  a  dollar^  liow 
many  of  each  kind  must  he  take? 

23.  A  has  a  sum  of  money  invested  at  a  cerlaiii  rate  of 
interest,  li  has  ^loOO  more  invested,  at  a  rate  1  per  cent, 
hinher,  and  thus  <j:ains  $S()  more  interest  than  A.  C  has  in- 
vested ^^7^{)()  more  than  15,  at  a  I'ate  still  hi^dier  hy  1  [)er  cent,, 
and  thus  Ljains  $70  more  than  H.  What  is  the  amount  each 
l)erson  has  invested  and  the  rate  of  interest  r* 

24.  A  <i^r()cei-  had  three  casks  of  wine,  coidainini^  in  all 
344  gallons,  lie  sells  ")()  ^falloiis  iVom  the  first  cask;  then 
])ours  into  the  first  one-third  of  what  is  in  the  second,  and 
then  into  the  secoiul  oni'-til'th  of  what  is  in  the  third,  after 
which  the  first  contains  10  gallons  more  than  the  second, 
and  the  second  10  more  than  the  third.  How  v.nuAx  wine  did 
each  cask  contain  at  first  ? 


o  fro 

Ui>S- 

They 
from 
tis  the 

>e  for 

d  the 

what 


V 


Equivalent  iiiul  Inconsistent  Equations. 

111.  It  is  not  always  tlie  case  tliat  values  of  two  unknown 
quantities  can  l)e  found  from  two  eejuations.  If,  for  example, 
A\  e  have  the  etjuations 

X  4-  ^>//  r-  ;], 

j).f  +  4'//  =  G, 

we  sec  that  the  scc(;nd  can  be  derived  from  the  first  by  multi- 

plyini^  both  members  by  2.     llcnce  every  pair  of  values  of  x 

ami  II  which  s:itisfy  the  one  will  satisfy  the  other  also,  so  that 

the  two  are  ecpiivalent  to  a  single  one. 

If  the  cqiuitions  were 

a;  +  Sy  =  5, 

2x  +  42/  =  G, 

there  wonld  be  no  values  of  x  and  y  which  wonld  satisfy  both 
e(  [nations. 


122 


EQUATIONS  OF   TUB  FIRST  DEGREE. 


For,  if  \ve  multiply  the  first  by  2  and  subtract  the  second 
from  the  product,  we  shall  have, 

2a;  4-  4y  =  10 


2ri;  +  4?/  =    6 


1st  eq.  X  2, 

2d  cq.,  

Remainder,  0=4, 

an  impossible  result,  which  shows  that  the  equations  are  incon- 
sistent. This  will  be  evident  from  the  equations  themselves, 
because  every  pair  of  values  of  x  and  y  which  gives 

2a;  +  4y  =  6, 

must  also  give  a;  +  2?/  =  3, 

and  therefore  cannot  give   a;  +  2?/  =  5. 

143.  Generalization  of  the  preceding  result.  If  we  take 
any  two  equations  of  the  first  degree  between  x  and  y  which 
we  may  represent  in  the  form 

ax  +  hy  =  c,  \  ,  . 

a'x  +  b'y  =  c',  )  ^^' 

and  eliminate  x  by  addition  or  subtraction,  as  in  §  137,  we  have 
for  the  equation  in  y, 

{a'b  —  ah)  y  =z  a'c  —  ac'. 

Now  it  may  happen  that  we  have, 

a'l)  —  ab'  =  0  identically.  (2) 

In  this  case  y  will  disappear  as  well  as  x,  and  the  result 

will  be 

a'c  —  ac'  =  0. 

If  this  equation  is  identically  true,  the  two  equations  (1) 
will  be  equivalent ;  if  not  true,  they  will  be  inconsistent.  In 
neither  case  can  we  derive  any  value  of  y  or  x. 

If  we  divide  the  above  equation,  (2),  by  aa'  we  shall  have 

b  _  5'^ 

a  ~  a'' 
Hence, 

TJieorem.  If  the  quoti  ;nt  of  the  coefficients  of  the 
unknown  quantities  is  the  same  in  the  two  equations, 
they  will  be  either  equivalent  or  inconsistent. 


I 


L 


VJ 

ti. 


SI 
01 
0. 


INEQUALITIES. 


123 


This  theorem  can  be  expressed  in  the  following  form : 

//  the  terms  containing  the  unknown  quantity  in  the 
one  equation  can  he  multiplied  by  such  a  factor  that 
they  shall  hotli  become  equal  to  the  corresponding  terms 
of  the  other  equation,  the  two  equations  will  be  either 
equivalent  or  inconsistent. 

Proof.  If  there  be  such  a  factor  m  that  multiplying  the 
first  equation  (1)  by  it,  we  shall  have 

ma  =  a' J 
mb  =  b'. 
Eliminating  m,  we  find 

a'b  —  ab'  =  0, 
the  criterion  of  inconsistency  or  equivalence. 

143.  When  two  equations  are  inconsistent,  there  are  no 
values  of  the  unknown  quantities  which  will  satisfy  both  equa- 
tions. 

When  they  are  equivalent,  it  is  the  same  as  if  we  had  a 
single  equation  ;  that  is,  we  may  assign  any  value  we  please  to 
one  of  the  unknown  quantities,  and  find  a  corresponding  value 
of  the  other. 


-♦"♦-♦- 


IS  (1) 
:.     In 

have 


)f  the 
Itions, 


CHAPTER     IV. 

OF     I  NEQUALITIES. 

144.  Def.  An  Inequality  is  a  statement,  in  the 
language  of  Algebra,  that  one  quantity  is  algebraically 
greater  or  less  than  another, 

Def.  The  quantities  declared  unequal  are  called 
Members  of  the  inequality. 

The  statement  that  A  is  greater  than  B,  or  that  ^  —  ^  is 

positive,  is  expressed  by 

A>  B. 


(I 


124 


INEQUALITIES. 


'» 


That  A  is  less  thtin  B,  or  that  ^1  —  i>  is  negative  is 
cxi)icsse(l  by 

A  <  /?. 

The  form  .1  >  />'  >  C 

indicates  that  the  (juantity  B  is  less  than  A  but  greater  tliau  (J, 

The  form  A^  B 

indicates  tliat  A  may  bo  either  equal  to  or  greater  than  B,  but 
cannot  be  less  than  B. 

Properties  of  Inequalities. 

145.  TJieoTcm  I.  An  incHxnality  will  still  suosist 
aft(T  tlu^  sanio  quantity  lias  been  added  to  or  subtracted 
Ironi  each  member. 

Proof.  If  the  incf[uahty  be  A  y  B,  A  —  B  must  be  posi- 
tive. If  wo  add  the  same  quantity  //to  A  and  B,  or  subtract 
it  from  them,  we  shall  have  ^1  i  ^/ — (/>±-^^)>  which  is 
equal  to  .1  —  B,  and  therefore  positive.     Hence,  if 

A  >  /;, 

tlien  A±II>  B  ±  II. 

Cor.  If  any  term  of  an  inequality  be  transposed 
and  its  sign  changed,  the  inequality  will  remain  true. 

TJicoreni  II.  An  inequality  will  still  subsist  after 
its  members  have  been  multiplied  or  divided  by  the 
same  positive  number. 

Proof.  If  ^  —  B  is  positive,  then  {m  or  ??,  being  positive) 
m  {A  —  B)  or  mA  —  mB  will  be  positive,  and  so  will 


A-B           A 

or 

Qi                 n 

B 

n 

Hence,  if 

A>  B, 

then 
and 

771 A  >  VI B, 

n       n 

' 


,■  ^ 


I, 


. 


V 


INEQUALITIES. 


125 


It  may  bo  shown  in  the  same  way  that  if  m  or  n  is  negative, 
mA  —  7nB  or will  he  nerativc.     ITeiice, 

Tlieorcm  III.  If  both  members  of  an  inequality  bo 
multiplied  or  divided  by  the  same  negative  number, 
the  direction  of  tlie  inequality  will  be  reversed. 

That  is,  if  A  >  B,      • 

then  ~  7nA  <  —  mB, 


and 


n 


< 


B 

— -  • 

n 


TJicorem  IV.  If  the  corresponding  members  of 
several  inequalities  be  added,  the  sum  of  the  greater 
members  will  exceed  the  sum  of  the  lesser  members. 

Tlieore,  >  r  If  the  members  of  one  inequality  be 
subtracted  from  the  non-corresponding  members  of 
anotluM',  the  inequality  will  still  subt>ist  in  the  direction 
of  tlie  latter. 

That  is,  if  Ay  B, 

then  A  —  y  y  B  —  x. 

The  proof  of  the  last  three  theorems  is  so  simple  that  it  may  he  sup- 
plied by  the  student. 

Theorem  VI.  If  two  positive  members  of  an  in- 
equality be  raised  to  any  power,  the  inequality  will 
still  subsist  in  the  same  direction. 

Proof.     Let  the  ineqnality  he 

A>  B.  (a) 

Becanse  A  is  positive,  we  i^hall  have,  by  multiplying  by  A 
(Th.  II), 

A^  >  AB.  (1) 

Also,  because  B  is  positive,  wc  have,  by  multiplying  (a) 

hy  B, 

AB  >  /A  {:l) 


'P 


126 


INEQUALITIES. 


Therefore,  from  (1)  and  (2), 

A'^yB^,  (3) 

Multiplying  the  last  inequality  by  A, 

A^  >  A  IP.  (4) 

Multiplying  (3)  by  i?, 

AJP  >  B^.  (5) 

Whence,  A^  >  B^. 

The  process  may  be  continued  to  any  extent. 

Examples  of  the  Use  of  Inequalities. 

14G.    Ex.  I.  If  a  and  b  be  two  positive  quantities,  such 
that 

■we  must  have  a  +  Z»  >  1. 

Proof.    If  a-\-i^l, 

we  should  have,  by  squaring  the  members  (Th,  YI), 

a2  +  2aJ  +  ^2  =  1 . 

and  by  transposing  the  product  2ab  (Th.  I,  Cor.), 

a2  +  52  ^  1  _  2ab. 

Because  a  and  b  are  positive,  2ab  is  positive,  and   * 

l  —  2ab<  1. 
Therefore  we  should  have 

««  +  *2  <  1, 

and  could  not  have  0,^  +  1^=1^  as  was  originally  supposed. 
Ex.  2.    If  a,  b,  m,  and  n  are  positive  quantities,  such  that 


am 
b^  n' 


(a) 


then  the  value  of  the  fraction will  be  contained  between 

a  4-  w 

{I  711 

the  values  of  ,  and  —  ;   that  is, 
b  n 


w 


ze 


th< 


• 
I 


INEQUALITIES. 


127 


een 


'i 


(1) 


(2) 


(3) 


a      a  -^  m       m 

To  prove  the  first  inequality,  we  must  show  that 

a      a  -\-  m 
b  ~~  V+li 

is  positive.     Reducing  this  expression  by  §  lOG,  it  becomes 

an  —  hn 

bJbT^' 

From  the  original  inequality  (a)  we  have,  by  multiplying 
by  the  positive  factor  biif 

a?i  y  bm. 

That  is,  a7i  —  5m  is  positive  ;  therefore  the  fraction  (3) 
with  this  positive  numerator  is  also  positive,  and  (2)  is  positive 
ae  asserted. 

The  second  inequality  (1)  may  be  proved  in  the  same  way. 

EXERCISES. 

I.  Prove  that  if  a  and  b  be  any  quantities  diflferent  from 
zero,  and  1  >  a;  >  —  1,  we  must  have 

a2  —  2abx  +  i^  >  0. 


2.  Prove  that  y^-j-)  >  «^- 

3.  U  dx  —  5>  13,  then  x  >  6. 


Sx 


4.  li  Gx>~-\-  18,  then  a;  >  4. 

5.  If  y-f  >|-3,  thenx>5. 

6.  If  fit  —  nxyp  —  qx,   then   x  >   -^—  • 


'  m  y 


and  m  is  positive,  then  x  <C  y. 


8.  If  «2  -f  52  ^  ^2  =:  1,  and  a,  b,  and  c  are  not  all  equal, 

then  ab  ■\- be  +  ca  <.  1. 

Sr«<3EBTi0N.    The  B(iuares  of   a  —  h,h  —  c,   and  c  —  a  cannot  be 
nregalive. 


BOOK    IV. 
A'  ATIO    A  ND    PR  OPO R  PI O N. 


CHAPTER    I. 
NATURE    OF     A     RATIO. 

147.  Drf.  The  Ratio  of  a  quantity  A  to  another 
quantity  B  is  a  number  expressing  the  value  of  A  wlien 
com])ared  with  B  as  the  standard  or  unit  of  measure. 

Examples.  Comparing 
the  lengths  A,  B,  C,  D,  it 
■will  be  seen  that 

A  is  2|-  times  D\ 

^  is  ^  of  D', 

C  is  I  of  D. 


A 
li 
C 
J) 


I     I     I     I     I     I 


I     I     I     I     I 


Wc  express  this  relation  hy  saying. 


to' 


The  ratio  oi  A  to  D  is  2^  or 


0 

_  t 

'4' 


t< 


ft 


"     B  to  D  is 


2 


(1) 


{( 


C  to  i)  is  7- 
4 


1-18.  The  ratio  of  one  quantity  to  another  is  expressed  hy 
writing  the  unit  of  measure  after  the  quantity  measured,  and 
inserting  a  colon  between  them. 

The  statements  (1)  will  then  he  expressed  thus  : 


A:D  =  2} 


0 
4' 


B:D=: 


C'.D 


3 

4" 


Dff.    The  two  quantities  compared  to  form  a  ratio 
are  called  its  Terms. 


"1 


RATIO. 


\'>[) 


tlier 
lien 
e. 


0) 


Jl.y 
and 


1 


atio 


Dcf.  The  qiuiiitity  nu'asured,  or  the  liist  term  ol' 
the  ]*iitio,  is  ciilh'd  the  Antecedent. 

The  unit  of  lueiisure,  or  tlie  second  term  of  the  ratio, 
is  called  the  Consequent. 

Rem.  When  tlio  antecedent  is  greater  than  the  conse(iuentj 
the  ratio  is  greater  than  nnity. 

When  the  antecedent  is  Itos  tlum  the  consequent,  the  ratio 
is  less  than  unity. 

141).  To  find  the  ratio  of  a  quantity  yl  to  a  standard  U, 
■\ve  imagine  ourselves  as  measuring  oil:  the  quantity  A  witli  6''as 
a  cari)enter  meiisurcs  a  hoard  with  his  foot-rule. 

There  are  then  tlirec  cases  to  he  considered,  according  to 
the  wav  the  measures  come  out. 

Case  I.  AVe  may  find  tluit,  at  tlic  end,  A  comes  out  an 
exact  nunil)er  of  times  V.  The  ratio  is  then  a  whole  numljer, 
and  Ave  say  that  U  exactly  measures  A,  or  tiiat  A  is  a 
multiple  of  U. 

Case  II.  We  may  find  that,  at  the  end,  the  measure  docs 
not  come  out  exact,  hut  a  i)iccc  of  A  less  than  U  is  left  over. 
Or,  A  may  itself  he  less  than  U.  We  must  then  fiiul  what 
fraction  of  U  the  piece  left  over  is  equal  to.  This  is  done  hy 
dividing  U  up  into  such  a  numher  of  equal  parts  that  one  of 
these  parts  shall  exactly  measure  A  or  the  piece  of  A  which  is 
left  over.  The  ratio  will  then  ho  a  fraction  of  Avhich  the  num- 
her of  parts  into  which  U  is  divided  will  ho  the  denominator, 
and  the  numher  of  these  parts  in  A  the  numerator. 

Example.    If  we  find  that  , 

hy  dividing  U  into  7  parts,  4  of 
these  parts  will  exactly  make  A, 


I     \     I 


1  =  A 


then  A  =  4"  <^f  U,  and  Ave  have  for  the  ratio  of  A  to  U, 

A:U=t 

If  we  find  that  A  contains  U  3  times,  and  that  there  is 
then  a  piece  equal  to  -\  of  U  left  over,  Ave  have 


A     '.U  =^=    y 


i!l 


-  m 


9 


130 


RATIO. 


'I 


The  3  C/''s  arc  equal  to  ^  of  U,  so  that  wc  may  also  say 

A 


A=^oiU, 


or 


u  =  f . 


Avhicli  i?  simply  the  result  of  reducing  the  ratio  3^  to  an  im- 
l)ro])or  Iriietion. 

In  general,  if  we  find  that  by  dividing  U  into  n  parts,  -I 
■will  be  exactly  m  of  these  parts,  then 

A  '.  U  =  —, 

n 

•whether  m  is  greater  or  less  than  n. 

When  the  magnitude  of  A  measured  by  U  can  be  exactly 
expressed  by  a  vulgar  fraction,  A  and  U  are  said  to  be  com- 
mensurable. 

Case  III.  It  may  happen  that  there  is  no  number  or  frac- 
tion which  will  exactly  express  the  ratio  of  the  two  magnitudes. 
The  latter  are  then  said  to  be  incommensurable. 

150.  Theorem.  The  ratio  of  two  incommensurable 
magnitudes  may  always  be  expressed  as  near  the  true 
value  as  we  please  by  means  of  a  fraction,  if  we  only 
make  the  denominator  large  enough. 

Examples.  Let  us  divide  the  unit  of  measure  into  20 
parts,  and  suppose  that  the  antecedent  contains  more  than  28 
but  less  than  29  of  these  parts.  Then,  by  supposing  it  to  con- 
tain 28  i)arts,  the  limit  of  error  will  be  one  part,  or  ^^  of  the 
standard  unit. 

In  general,  if  we  wish  to  express  the  ratio  within  1  n^h  of 
the  unit,  we  can  certainly  do  it  by  dividing  the  unit  into  n  or 
more  parts,  or  by  taking  as  the  denominator  of  the  fraction  a 
number  not  less  than  n. 

llludration  hy  Decimal  Fractions.  The  square  root  of  2 
cannot  be  rigorously  expressed  as  a  vulgar  or  decimal  fraction. 
But,  if  we  suppose 

Vs  =  1.4      =  i^,    the  error  will  be  <  ^V  5 

^2  =  1.41    =\U,      "  "     <tU; 

V2  =  1.414  =  HU,     "  "      <ToW- 

etc.        etc.  etc.  etc. 


NATURE  OF  A   RATIO. 


181 


20 


V. 


^  \. 


Since  the  decimals  may  be  continued  without  end,  tlio 
square  root  of  2  can  be  expressed  as  a  decimal  fraction  wi(li  an 
error  less  tlian  any  assignable  (quantity.  This  general  fact  is 
expressed  by  saying : 

Tlie  Umib  of  the  ciTor  which  ive  make  by  representing 
an  iiicmnincnsiu'uhlo  ratio  as  a  fraction'  is  zero. 

151.  Batio  Hfi  a  Quotient.  From  Ciise  II  and  the  explana- 
tions wliieli  i)recedc  it  we  see  that  when  we  say 

wc  mean  the  same  thing  as  if  we  had  said, 

A  is  I  of  C7,    or    A  =  \U. 

If  A  and  U  are  nmnbcrSy  wc  may  divide  both  sides  of  this 
equation  by  U,  and  obtain, 

^  _  4 

J/  "  7*  . 

"We  therefore  conclude  that  when  A  and  U  arc  numbers. 

That  IS,  (/ 

Tlieorem.  The  ratio  of  two  numbers  is  equal  to  the 
quotient  obtained  by  dividing  the  antecedent  term  by 
tlie  consequent. 

In  the  case  of  magnitudes,  the  relation  of  a  ratio  to  a  quo- 
tient may  be  shown  thus  : 

Let  us  have  two  magnitucics  M  and  F,  such  that  M  is 
4  times  V.     Then  we  may  write  the  relation, 

J/=4F. 

Dividing  by  4,  we  have 


M 


:=   V. 


Since  V  is  not  a  number,  we  cannot,  strictly  speaking, 
multiply  or  divide  by  it.  But  we  may  take  the  ratio  of  M  to 
F  without  regard  to  number,  a.id  thus  find, 

M  :  F  =  4. 


I  i 


!l{l 


I  I 


ii 


ii 


132 


It  AT  10. 


u 


Iii:m.  Tlio  tlioory  of  ralioH  llic  (cniis  of  wliich  arc  imijjni- 
liidcs  and  not  niunlH'i's,  is  treated  in  (iconietry. 

In  Ai<,'el)ni  we  consider  t lie  ratios  of  numbers,  or  of  nia^'- 
nitudes  represented  by  niinil)ers. 

15'^.  Dcf.  If  w(»  iiitvrchungi^  tlio  teriua  of  u  ratio, 
the  result  is  called  the  Inverse  ratio. 

That  is,    U :  A  is  tlio  inverse  of  A  \  U. 


If 

then 


U  :  A  =  —. 

n 


U 


m 
n 


A, 


and  wc  have,  by  dividing  by  — , 

A  = 

or  A  :  U  = 


n 
m 
n 
m 


r^. 


Because  —  is  the  reciprocal  of      ,  wo  conclude : 

TJii'orcin.  The  inverse  ratio  is  the  reciprocal  of  the 
direct  ratio. 

Properties  of  Ratios. 

1  i*.*?.  TJieorem  I.  If  both  terms  of  a  ratio  be  multi- 
plied by  the  same  factor  or  divided  by  the  same  divisor, 
the  ratio  is  not  altered. 


Proof.     Ratio  of  B  to  A  =  B  \  A  = 
If  m  be  the  ftictor,  then 


/>' 


7nB       B 
Eatio  of  mB  to  mA  =  mB  :  mA  =  — -j  =  -?> 

mA       A 

the  same  as  the  ratio  of  i?  to  ^1. 

154.  Theorem  II.     If  both  terms  of  a  ratio  be  in- 
creased by  the  same  quantity,  the  ratio  will  be  increased 


1 


riioponrioy. 


i\V6 


ifil  is  less  tli.'in  1,  niid  (rmiiiiislird  il'lt  ls;j;ivaU'i' than  I  ; 
tliat  is,  it  will  be  broiiglit  nv.nvv  to  unity. 


RvAMl'l.K.     Let  tlioorl^iniil  ratio  be  2  :  5 


If  we  rrpratnllv  luM 


1  to  Itotii  iiimicmtor  ami  druuiuiiiulur  uf  ihu  fmctiuu,  wc  hhull  Iiuvd  tlio 
BcricH  of  I'mctiuiiH, 

>      tf»       7'       N»      ^*»"» 

each  of  wliich  is  grcator  tlmn  the  pri'cediii},'.  becnuso 
*  -  s  =  A  ;       wlicncc,     «  >  «. 


^ 


3  _    a  . 

11  —  \i  ' 

*    _    I    !»     . 


ulxMicc,     i  >  i}. 
wlu'iicc,     5  >  7. 


etc.  etc. 

General  J'rnnf.  Lot  // :  Z*  be  tlic  ori^^iiial  ratio,  and  let 
both  tiTiiis  bo  increased  l)y  the  quantity  //,  niakiii;^'  the  now 
ratio  «+  u  :  0-\-u.     The  now  ratio  viiniis  tlic  old  one  will  bo 

{b  —  ft)  u 

If  h  is  greater  than  a,  this  (|iiaiitity  Avill  be  ]>ositive,  show- 
ing tliat  the  ratio  i-  increased  l)y  adding  n.  It'/'  is  icss  tlian  a, 
tlie  ([uantity  will  l»c  negative,  showing  that  the  ratio  is  dimin- 
ished by  adding  u. 


-♦■♦♦- 


■r 


I 


CHAPTER    II. 

PROPORTION. 

1,15.  Dif.    Proportion  is  an  equality  of  two  or 
more  ratios. 

Since  each  ratio  has  two  terms,  a  proportion  must  have  at 
least  four  terms. 

D<f.    Tlu^  forms  wliicli  enter  into  two  equal  ratios 
are  called  Terms  of  the  proportion. 

1(  a  :  b  bo  one  of  the  ratios,  and  jj  ;  q  the  other,  the  pro- 
portion will  be, 

a  :  b  =  p  :  q.  (1) 


i''^;!ii! 


]i 


•It 


134 


PROPORTION. 


A  proportion  is  sometimes  written, 

a  :  b    : :   p  '.  q, 

which  is  read,  "  As  a  is  to  h  so  is  p  to  q."  The  first  form  is  to  be  pre- 
ferred, because  no  other  sign  than  that  of  equality  is  necessary,  but  the 
equation  may  be  read,  "  As  a  is  to  6  so  is  p  to  q"  whenever  that  expres- 
sion is  the  clearer. 

Def.  Tlie  first  and  fourtli  terms  of  a  proportion  are 
called  tli(;  Extreme.?,  the  second  and  third  are  called 
the  Me£ais. 

Theorems  of  Proportion. 

150.  TJieorem  I.  In  a  proportion  the  product  of 
the  extremes  is  equal  to  the  product  of  the  means. 

Proof.  Let  us  w.-'te  the  ratios  in  the  proportion  (1)  in  the 
form  of  fractions.     It  will  give  the  equation, 


a  _p 


(3) 


Multiplying  both  sides  of  this  equation  by  Iq,  we  shall  have 

(tr  =  hp.  (3) 

Cor.  If  there  fi  re  two  unknown  terms  in  a  propor- 
tion, they  may  l^e  exp  essed  by  a  single  unknown 
symbol. 

Example.    If  it  be  required  that  one  quantity  shall  be  to 

another  as  p  to  q,  we  may  call  the  first  px  and  the  second  qx, 

because 

px  :  qx  zsi  p  '.  q  (identically). 

157.  Theorem  II.  If  the  means  in  a  proportion  be 
inter(!hanged,  the  proportion  will  still  be  true. 

PrQof.  Divide  the  equation  (3)  by  j^Q.'  We  shall  then 
liave,  instead  of  the  proportion  (1), 

a_h 
p~  q' 
or  a  \  p  =.  h  '.  q. 


PROPORTION. 


135 


\ 


Def.  The  proportion  in  which  the  means  are  inter- 
changed is  called  the  Alternate  of  the  original  pro- 
portion. 

The  following  examples  of  alternate  proportions  should  be  studied, 
and  the  truth  of  the  equations  proved  by  calculation  : 

1:2=    4:8;       alteniate,     1:4    =2:8. 
2:3=G:9;  "  2:0=3:9. 

5  :  2  =  25  :  10  ;  "  5  :  25  =  2  :  10. 

158.  TJieorem  III.  If,  in  a  proportion,  we  increase 
or  diminish  each  antecedent  hy  its  consequent,  or  each 
consi'quent  by  its  own  antecedent,  the  proportion  will 
still  be  true. 

Example.    In  the  proportion, 

5  :  2  =  25  :  10, 

the  antecedents  are  5  and  25,  the  consequents  2  and  10  (§  148).     Increasing 
eacl'i  antecedent  by  its  own  consequent,  the  proportion  will  be 

5  +  2:2  =  25  +  10:10,        or        7:2^35:10. 
Diminishing  each  antecedent  by  its  consequent,  the  proportion  will 

U6COIIl(3 

'  5  -  2  :  2  =  25  -  10  :  10,        or        3  :  2  =  15  :  10. 
Increasing  each  consequent  by  its  antecedent,  the  jtroportion  will  be 

5  :  2  +  5  -  25  :  10  +  25,        or        5  :  7  =  25  :  35. 
These  equations  are  all  to  be  proved  numerically. 

General  Proof.     Let  us  put  the  proportion  in  the  f(  rm 

b       q     .  ^  ' 

If  we  add  1  to  eacli  side  of  this  equation  and  reduce  each 
side,  it  will  give 

b       ~       q      ' 
that  is,  a  -\-  b  :  b  =  p  -{-  q  :  q.  (5) 

In  the  same  way,  by  subtracting  1  from  cacli  side,  it  will  he 
a  ~b  ',  h  =■  })  —  q  '.  q.  (0) 


'■§.. 

Mai 


130 


PItOPOllTION. 


If  we  invert  tlie  fractions  in  equation  (4),  the  liittor  will 


become 


a 


P 


M 


By  adding  or  subtracting  1  from  each  side  of  tliis  equation, 

and  then  again  inverting  tlie  terms  of  tlie  reduced  fractions, 

we  shall  lind, 

a  :  h  -\-  a  =:  2^  :  q  -\-  2^ ; 

a  :  b  —  a  =^  p  :  q  —  2^' 

The  form  (5)  was  formerly  designatod  as  formed  "  by  composition," 
and  (0)  as  formed  "  by  division."  But  these  terms  are  now  useless,  be- 
cause all  the  above  forms  are  only  special  cases  of  a  more  general  one  to 
be  now  explained. 

IVSO.  Theorem  IV.  If  four  quantities  form  the  pro- 
portion 

a  :  h  =  c  :  (7,  (a) 

and  if  m,  n,  ^;,  and  q  be  any  multipliers  whatever,  we 
shall  have 

ma  +  nh  :  ^y«  -\-  qh  —  mc  +  nd  :  ^:>c  +  qd. 
Proof.     The  proportion  {(()  gives  the  equation. 


a  _  c 

b  ~  d 


V 


Multiplying   this  ecpiation   by    -   and   adding   1   to  each 


member, 


qb  +  ^  -  qd  +  ^• 


Itedueing  eaeli  member  to  a  fraction   and   inverting   the 

terms, 

qb       __    jjd 

2m  4-  ([I)  ~  pc  -\-  qd 
Dividing  both  members  by  q, 

2m  ■\-  qb       pc  +  (jd  ^  ' 

The  original  proportion  {a)  also  gives,  by  inversion, 


an 
i\n 
of 


PROPORTION. 


137 


et 


ach 


ig   the 


(^) 


a  ~~  c* 
from  which  we  obtain,  by  nmUiplying  by  -,  addhig  1,  etc., 

(jb  4-  pfi       Q(^  +  pc 

2M  pc 

a  e 


])a  -\-  qb       ji^c  +  qd 

(8)  X  m  +  ( <)  X  n  gives  tlic  equation, 

ma  -\-  nb  inc  ■\-  nd 

pa  -\-  qb        pc  -{-  qd  * 

or  ma  -\-  nh  :  pa  -{-  qb  =  vie  -f  nd  :  21c  +  qd, 

which  is  the  result  to  be  demonstrated. 


C^) 


(9) 


KJO.  TV)  cor  em  V.  If  each  term  of  a  proportion  be 
rais<*d  to  the  same  power,  the  proportion  will  still 
subsist. 


Pi 

or 

oof. 

If 

a  :  b  - 
a 
b  ~ 

=  p  :  q, 
.P 

-q' 

then. 

bv 

mult 

iplying 

each  mcmljer 

by 

itself 

repe 

itedly, 

we 

shall  have 

«2 

^P\ 

• 

«3 

etc. 

etc. 

Hence,  in  general, 


Cor.  n 

then 
and 


fl»  :   h^  z=  pn   .  qn, 

a  :  b  =  p  '  q, 

an  :  a"  ±  b''  =  p"'  :  p'^  ±  r/«  ; 

f(n  _|_  /,n   .    //I  _  ^^n  -j_  f^n   .    ,j?i^ 

Til  cor  em  VI.  When  throe  terms  of  a  proportion 
are  given,  the  fourth  can  always  be  found  fi-om  the 
theorem  that  the  i)roduct  of  the  means  is  equal  to  that 
of  the  extremes. 


1.1 


ijillii 


ri^ 


138 


PROPORTION. 


We  hav 

e  shown  that  whenever 

a  :  b 

=  p  '.  q, 

lIlOil 

aq 

=z  bp. 

Considering  the  different  terms  in  succession 

as 

unknown 

quantities, 

wc  lind, 

a 

h 

hp 

-  t' 

_aq 

~~P' 
aq 

-   b' 

q  =z 


a 


Cor.  1.     If,  in  the  general  equation   of  tlie   fir^' 

degree 

ax  +  hy  =  c, 

the  term  c  vaiiislies,  the  equation  determmes  the  ratio 
of  tlie  unknown  quantities. 

Proof.     If  ax  -\-  by  =  0, 

then  ax  z=z  —  by, 

X  b 

and  -  = , 

y  a 

or  X  :  y  =  —  b  :  a. 

Cor.  2.  Conversely,  if  the  ratio  of  two  unknown 
quantities  is  given,  the  relation  between  them  may  be 
expressed  by  an  equation  of  the  iirst  degree. 


The  Mean  Proportional, 

IGl.  D(f.  When  the  middle  terms  of  a  proportion 
are  equal,  either  of  them  is  called  the  Mean  Propor- 
tional between  the  extremes. 

Tlic  fact  that  b  is  the  mean  proportional  between  a  and  c 
is  expressed  in  the  form, 

a  :  b  =  b  :  c. 


PROPORTION. 


139 


Theorem  I  tlien  gives,  b^  =  ac. 

Extracting  the  square  root  of  both  members,  wc  have 

b  =■  Vac. 
Hence, 

Tlieorem  VII.     The  mean  proportional  of  two  quan- 
tities is  equal  to  the  square  root  of  their  product. 


)rtion 
|opor- 

and  c 


Multiple  Proportions. 

1G3.  We  may  have  any  number  of  ratios  equal  to  each 
other,  as 

a  \  b  =  c  :  d  =^  e  '.  f,  etc. 

G  :  4  =  9  :  G  =  3  :  3  =  21  :  14.  {a) 

Such  proportions  are  sometimes  written  in  tlie  form 

6  :  9  :  3  :  21  =  4  :  G  :  2  :  14.  {b) 

In  the  form  {b)  the  antecedents  are  all  written  on  one  side 
of  the  equation,  and  tlie  consequents  on  the  other.  Any  two 
numbers  on  one  side  then  have  the  same  ratio  as  tlie  cor- 
responding two  on  the  other,  and  the  proportions  expressed  by 
this  equality  of  ratios  are  the  alternates  of  the  original  propor- 
tions (rt).     For  instance,  in  the  proportion  {b)  we  have, 


6:9    =4:6,  which  is  the  alternate  of  6  :  4  =    9 
6:3=4:2,"        "  "  6:4=3 

6  :  21  =  4  :  14,    "        "  "  6:4  =  21 


9  :  21  =  6  :  14, 


a 


i( 


It 


9  :  6  =  21 


6. 
2. 
14. 

14. 


163.  A  multiple  proportion  may  also  be  expressed  by  a 
number  of  equations  equal  to  that  of  the  ratios.    Since 

a  :  b  =  c  :  d  =  e  '•  f,  etc., 

let  U3  call  r  the  common  value  of  these  ratios,  so  that 


a 

—    —    7' 
b~      ' 


Then 


d 

a  =  rb, 
c  =  rdy 
e  —  rfy 


=  r,    etc. 


(^■) 


i1 


140 


PBOPORTIOX. 


w 


will  express  tlic  same  relations  between  the  c^uantities  a,  h,  c, 
d,  e,  f,  etc.,  that  is  (3xpre.<setl  hy 

a  :  b  =z  c  :  d  -=  e  :  ff  etc.,  {a) 

or                      a  '.  c  '.  e  :  etc.  =  b  :  d  :  f  -.  etc.  (/;) 

It  will  be  seen  that  where  r  enters  in  the  form  {<•)  there  is  one  more 
equation  than  in  the  first  form  {a).  [In  this  form  each  =  rci)resents  an 
etiuation.]  This  is  becaust;  the  additional  quantity  r  is  introduced,  by 
eliininaiiiig  which  wo  diniinish  the  number  of  e(juations  by  one,  as  in 
eliminating  an  unknown  quantity. 

1G4.  Tlieorem.     In  a  multiple  proportion,  the  snm 

of  any  number  of  the  antecedents  is  to  the  sum  of  the 

corresponding  consequents  as  any  one  antecedent  is  to 

its  consequent. 

2      0       10      1? 


Ex.    We  have 


Tlien 


30 


2  +  ()4-104-l*2 
5  + 15  +  2T+~i]0 

which  has  the  same  value  as  the  other  four  functions. 


General  Proof.     Let  A,  B,  C\  etc..  ho  the  antecedents,  and 
a,  b,  c,  etc.,  the  corresponding  consequents,  so  that 

A  \  a  =  B  '.  b  =  C  '.  €,  etc.  (1) 

Let  us  call  r  the  common  ratio  A  i  a^  B  :  b,  etc.,  so  that 

A  =  ray 

B  =  rby 

C  =  re. 

etc.     etc. 

Adding  these  equations,  we  have 

yl  +  7?  +  6'  +  etc.  =  r  {a  -{- b  -]-  c -i-  etc.), 

^  +  //  4-  C  4-  etc. 


or 


—  /•; 


a  -{-  b  -\-  c  +  etc. 

that  is,  the  ratio  .1 +  />  + T'-f  etc.  :  r(-f  6-fr  +  ctc.  is  equal  to 
r,  the  common  value  of  the  ratios  A  :  a.  B  :  b,  etc. 


it 


PROBLEMS, 


I.  A  nuip  of  a  country  is  made  on  a  scale  of  5  miles  to 
3  inches. 


PROPORTION. 


141 


{a) 


(1.)  What  will  be  the  length  of  8,  VI,  17,  20,  33  miles  on 
the  map? 

i'L)  How  nifiny  miles  will  be  represented  by  0,  8,  10,  20, 

21»  iljrljc.>  1)11  the  llUlj)  ? 

I»i:m.  1.     If  .V,  I/,  J,  //,  r  b(3  the  nMiuircd  spaces  on  the  maj),  we  .shall 

5  :  ;3  -  8  :  ;?■  -  13  :  ?/,  etc. 
If  a,  h,  r,  etc.,  he  tlio  rc(iuired  number  of  miles,  we  shall  have 
3  :  T)  =  Q  :  a  =  8  :  b  -  U  :  r,  etc. 

Rem.  2.  When  there  are  several  ratios  comi)are(l,  a.s  in  this  problem, 
it  will  be  more  convenient  to  take  the  inverse  of  the  common  ratio,  and 
maMjilr  the  antecedent  of  each  following''  ratio  by  it  to  obtain  the  conse- 
qntjjl     111  the  tirst  of  the  above  proportions  the  inverse  ratio  is  3,  and 

X  =  ^  of  8,     y  =  l  of  12,  etc. 
In  the  second,       a  =  -i;  of  G,     &  =  jj  of  8,  etc. 

2.  To  divide  a  aiven  quantify  A  into  throe  ])arts  Avhieh 
sham  l>e  i)roportioiial  to  tiie  given  quantities  «,  0,  c,  that  is, 
iiit*>  tlie  parts  x,  ij,  and  z,  sueh  tluit 

X  '.  a  ^  y  :  h  =z  z  \  c, 
or  X  :  y  \  z  =z  a  '.  h  :  c. 

SoLUTlOX.     By  Theorem  IV, 

X  y  z  _  X  -\-  y  -\-  z  A 

a  ~  b  ~  c  ~  a  -\-  b  +  c        a  +  b  -\-  c 
Therefore, 

((A  1)A  cA 

:.'    y 


X  = 


a  -{-  b 


a  -\-  b  -Y  6-' 


z  = 


a  -\-  b  -\-  c 


miles  to 


%.  r>ivi(le  102  into  three  parts  which  shall  be  proportional 
to  the  numbers  2,  4,  11. 

4.  Divide  1000  into  five  parts  which  shall  be  proportional 
to  t!ie  numbers  1,  2,  3,  4,  5. 

5.  Find  tAvo  fractions  Avhose  ratio  shall  be  that  of  «  :  5,  and 
wiio.'^e  ^um  sliall  be  1. 

6.  Wliat  two  numbers  arc  those  whose  ratio  is  that  of  7  :  3 
and  whdse  dili'erenec  is  24. 

7.  What  two  numbers  are  those  whose  ratio  is  m  :  n,  and 
whose  difference  is  unity  ? 

8.  Find  x  and  y  from  the  conditions, 

X  :  y  =z  a  '.  b, 

HX  —  h;  =r  rt  +  b. 


i' 


'I!  I 


,  liU 


mm 


142 


niOPOllTION. 


9.  Show  that  if 


"vvc  must  iilso  liave 


a  :  b  =  A  '.  B, 
c  :  d  =  C  \  D, 
ac  :  bd  =  AC  :  IID. 


10.  Having  given  x  =  ay,  lind  the  vuluc  of  ^-^-^-. 

X  ~'Zy 


II.  Ilav 

ing  given 

find  the  value  of 

x  +  y 
x  —  y 

12.  If 

a  '.  b  =  2^  \  q, 

prove 

a2  +  ^2  : 

a  +  b       ^         ^      P  +  'I 

and 

a"  +  i«  : 

a  +  b-^    ^  ^    '  p  +  q 

13.  If 

a  +  b-{- 

c  -\-  d       a  —  b  +  c  —  d 

a  -{-  b  — 

c  —  d  ~  a  —  b  —  c  +  d' 

show  that 

a  :  b  =  c  :  d. 

14.  A  year's  profits  were  divided  among  three  partners,  A, 
B,  and  C,  proportional  to  the  numbers  2,  3,  and  7.  If  C 
should  i)ay  B  $l:i56,  their  shares  would  be  equal.  What  was 
the  amount  divided  ? 

15.  In  a  first  year's  partnership  between  A  and  B,  A  had 
2  shares  and  B  had  5.  In  the  second  year,  A  had  3  and  B  had  4. 
In  the  second  year,  A's  i)rofits  were  $3200  greater  and  B's  were 
$1700  greater  than  they  were  the  first.  What  was  each  year's 
profits  ? 

16.  In  a  poultry  yard  there  are  7  chickens  to  every  2  ducks, 
and  3  ducks  to  every  2  geese.     How  many  geese  were  there  to 

,  every  42  chickens  ? 

17.  A  drover  started  with  a  herd  containing  4  hordes  to' 
every  9  cattle.     He  sold  148  horses  and  108  cattle,  and  then 
liad  1  horse  to  every  3  cattle.     How  many  horses  and  cattle 
had  he  at  first  ? 

18.  If  a  l)owl  of  punch  contains  a  parts  of  water  and  b 
parts  of  wine,  wliat  is  the  ratio  of  the  wine  to  the  whole 
punch  ?  What  is  the  ratio  of  the  water  ?  What  are  the  sums 
of  these  ratios  ? 


th( 


in( 
wi 


PIWPOIITIOK 


113 


ers,  A, 
If  0 

lut  was 

A  had 
had  4. 
Ts  wore 
year's 


lucks, 
here  to 


:  c 


or.«es  to 
id  then 
cattle 

icr  and  b 
■whole 
he  sums 


19.  One  in.crot  consists  of  equal  i)iir<s  of  ffohl  and  silver, 
while  another  has  two  parts  ot  *^t>ltl  to  one  of  silver.  11"  I 
conibiiie  equal  weights  IVom  these  intj^ots,  what  jtroportion  of 
the  coin])ound  will  he  f:j()kl  and  what  i)ro])ortion  silver? 

20.  What  will  he  the  jtroportions  if.  in  the  y)reeeding  proh- 
leni,  I  conihine  one  ounce  from  the  lirst  ingot  with  three  from 
the  second  ? 

21.  One  cask  contains  a  gallons  of  water  and  b  gallons  of 
alcohol,  while  another  contains  m  gallons  of  water  and  u  of 
alcolud.  If  I  draw  one  gallon  from  each  cask  and  mix  them, 
what  will  be  the  quantities  of  alcolud  and  water  ? 

22.  What  will  be  the  ratio  of  the  lifpiors  in  the  last  case,  if 
I  mix  two  ])arts  from  the  first  cask  with  one  from  the  second  ? 

23.  WHiat  will  it  be  if  I  mix  2^  parts  from  the  first  with  q 
parts  from  the  second  ? 

24.  A  goldsmith  has  two  ingots,  each  consisting  of  an  alloy 
of  gold  and  silver.  If  he  comlnnes  two  ])arts  from  the  first 
ingot  with  one  from  the  second,  he  will  have  ecpial  parts  of 
gold  and  silver.  If  he  combines  one  part  from  the  first  with 
two  from  the  second,  he  will  have  3  parts  of  gold  to  5  of  silver. 
What  is  the  composition  of  each  ingot  ? 

SuoGESTiox.  Calf  r  tlie  ratio  of  the  wei^lit  of  pold  in  the  first  ingot 
to  the  whole  weiglit  of  tlie  ingot ;  then  1—7'  will  bo  tlie  ratio  of  the  sil- 
ver in  the  first  to  the  wliole  weight  of  the  ingot.  See  tlie  following 
question. 

Note.  Problems  iH-24  form  a  graduated  series,  introductory  to  the 
processes  of  Problem  24. 

25.  Point  out  the  mistake  which  would  be  made  if  the 
solution  of  the  preceding  problem  were  commenced  in  the  fol- 
lowing way  : 

If  the  first  ingot  contains  p  parts  of  gold  to  q  parts  of  silver,  and  the 
second  contains  ?•  parts  of  gold  to  s  of  silver,  then 

Two  parts  from  the  first  ingot  will  have  2;)  of  gold  and  2g  of  silver. 

One  part  from  the  second  ingot  will  have  r  of  gold  and  .<?  of  silver. 

Therefore,  the  combination  will  contain  2p  +  r  parts  of  gold,  and 
2^  +  8  parts  of  silver. 

Show  also  that  if  we  subject  p,  q,  r,  and  s  to  the  condition 

p  +  q  =  r  +  s, 
the  process  would  be  correct. 

26.  Show  that  if  the  second  term  of  a  proportion  lie  a 
mean  proportional  between  the  third  and  foiirtli,  the  third 
will  be  a  mean  proportional  between  the  first  and  second. 


^'1 


1  il 


\    4\\ 
1    ■ 

'  'i 

<  ^ 

, 

! 

!- 

! 

BOOK    V. 
OF    PO  WE R  S    A  ND    R  O  O  TS\ 


CHAPTER     I. 

INVOLUTION. 


Case  I.   Involution  of  Products  and  Quotients. 

1<>5.  Def.  The  result  of  takiiiii;  a  (luautity,  A^ 
n  times  as  a  factor  is  called  the  a/"'  power  of  A,  and 
as  already  kno^vIl  may  be  written  either 

AAA,  etc.,  n  times,     or    A'\ 

Def.  The  number  n  is  called  the  Index  of  the 
power. 

Def.  Involution  is  the  operation  of  finding  the 
powers  of  algebraic  eAjn'essions. 

Tlic  operation  of  invohition  may  always  he  expressed  hy 
ihe  apphcation  of  the  proper  exponcnl,  the  expression  to  be 
involved  being  inelosed  in  parentheses. 

Example.    The  n*''-  power  of  i  -{■  h  is  {a  +  ly. 
The  n*''-  power  of  ahc  is  {abcY. 

IGG.  Lwolution  of  Products.  The  n"'-  power  of  the 
product  of  several  factors  a,  h,  c,  may  bo  expressed  witliout 
exponents  as  follows: 

ahc  ahc  nlc,   etc., 

each  factor  being  repeated  n  times. 


tiej 


the 


exp( 


IS  m. 


ii 


INVOLUTION. 


145 


eiits. 
t,  luid 


of  the 

iig  tlie 

Used  l)y 
11  to  be 


of   the 
without 


IFore  Ihoro  will  he  iiUoirethor  n  a\  n  b\  air]  n  c's,  so 
that,  nsh\(i;  exponents,  tlie  whole  i)ower  will  be  «'*i"6'"  (g  00, 01). 

Hence,  {aOc)"'  =  a''b"c". 

Tiuit  is, 

Theorem.     Tlio  i)Ower  of  a  i)ioduct  is  equal  to  the 
product  of  the  powers  of  the  several  factors. 


1G7.  Inrohifion  of  Quotients.    Applying  the  same  methods 
to  fractions,  we  find  that  the  ?i"'  iiower  of  -  is  --  • 

y     r 


For 


(xY      X  X  X      , 
I  =  -  "     ,  etc.,  n  times  ; 
y'     y  y  y 

XXX,  etc.,  n  times 


ijyij^  etc.,  n  times 


(§  100) ; 


EXERCISES, 

Express  the  cubes  of 
I.     ttbc.  2. 


ah 

- — ■  • 

c 
a  —  h 


3.     abc~K 

,      mn  (a  +  b) 
o. 


pq  {a  —  b) 

Express  the  n^^  powers  of  the  same  quantities,  the  quanti- 
ties between  parentheses  being  treated  as  single  symbols. 

Case  II.    Involution  of  Powers. 

108.  Problem.     It  is  required  to  raise  the  quantity  a^  to 
the  n'^  i)ower. 

Solution.    The  n*^  power  of  a^  is,  by  definition, 

a"*  X  rt^  X  a*",  etc.,  n  times. 

By  §  GG,  the  exponents  of  a  are  all  to  be  added,  and  as  the 
exponent  m  is  repeated  n  times,  the  sum 

m  +  m  +  m  -f  etc.,  n  times, 

is  mn.     Hence  the  result  is  a'"",  or,  in  the  language  of  Algebra, 

(rt'")»  =r  «'«». 

10 


M 
^ 


m 


1. 


"ill 


r 


140 


INVOLUTION. 


Hence, 

TJieorcm,.  If  any  powor  of  a  quantity  is  itself  to  Ih^ 
raised  to  a  ])()wer,  the  indices  of  tiie  powers  must  be 
multiplied  together. 

EXAMPLES. 

Note.     It  will  ho  HO(>n  tlmt  this  ihooreni  coincidos  with  that  of  Case  I 
when  Jiny  of  tho  fuctora  havo  tho  exponent  unity  undtTHtood. 

EXEHCISES. 

Write  the  cubes  of  the  following  (lUJintitics: 


I.     3ri/2. 
4.     hx*. 


2. 


,m 


5.     "Zahn^ 


3.     w 


6. 


"Write  the  ?i'^  jjowers  of 
7.     a.  8.     (fib. 

10.     (i^a^,  II.     'Zp^hf. 

14.     l{a  +  b  —  c){(i  —  h)P. 

Ans.  7"  {f(  -h  h  —  c)»  {a  —  h)>'P. 


9.     aWc, 
12.     {(t  +  h)  (c-hd). 


IS 


a 

•  r 


16. 


17' 


X  —  // 
Ans.  —  „  „„- • 


iS.     — ^. 
^^ 

ff^  (c  -  (If 

Reduce : 

20.     i^ZaWn^y.  21.     {—^mnx^y. 

22.     2rt(— ;Ji2/„;i3)3.  23.     (7;yr^)4. 

24.     (rtZ/«)*.  25.     (2«2,;3)«.  26.     {m^y. 

Note  1.  If  the  student  find  any  of  these  exponential  exiiressions 
difficult  of  expression,  he  may  lirst  express  them  by  writing  each  quantity 
a  number  of  times  indicated  by  its  exponent. 

Note  2.  The  student  is  expected  to  treat  the  quantities  in  paren- 
theses as  single  symbols. 


exp(  1 


INVOLUTTOK 


147 


t  bo 


Case  I 


(c  +  ^0- 


hyp. 


IproKsions 
1  (luuntity 


liu  parcn- 


llv.u.  Tho  procodiiif?  tlioorom  limls  ;i  practiciil  iipplicalioii 
"vvlu'ii  it  is  noc't'ssary  to  raise  a  small  niimlHT  (o  a  high  ixiwir. 
If,  lor  t'.\aini>k',  \vc  iiavo  to  raise  'i  to  the  .'JOtli  power,  we 
jshoiiltl,  willutut  this  theoivni,  have  to  iiiultiply  by  '4  no  less 
than  '>i\)  times,     lint  we  may  also  proceed  thus: 

2»   =  4, 

2*  =  y«.22  =  4-4        =  IC, 

29  =  a^.2«  =:  KJ.iG    =  )ir>r,, 

2"  =  2».28   =  25';'^       =  Vib'j'M'i, 

Case  ol*  Xc'jjative  ExpoiiciitH. 

1(»9.  Tho  profiHlinfT  theorom  may  he  a])))li('(l  to  nogalivc 
exponents.     J>y  the  delinition  of  sneh  exponents, 


liaising  the  Qrst  member  to  the  w'^*  power,  we  have, 


(1) 


a'lp 


=  a'^vb  "?. 


This  is  the  same  result  we  should  get  by  a))plyin,2f  tho 
theorem  to  the  second  member  of  (1),  and  i)rove,s  the  proposi. 
tion. 

EXERCISES. 


Express  the  Gth  powers  of 


lU 


'.-1 


a^h-\ 


3.     amp  •*. 

5.     (a^l>y{a-V)- 


a 


-m  7.-71 


6.     {x  +  ijY  {x  +  z) 


-n 


7. 


{a-b) 


tn 


Ticduce 


9.     [{a  +  7>)-i  {a  -  b)Y.       10.     (r/i-i6--2) 


I.     {ab'^c-'^y 


13. 


t}i-i\-i 


[xhj-i) 


12. 

14. 


{mhrJyK 


After  forming  the  expressions,  write  them  all  with  positive 
exponents,  in  the  form  of  fractions. 


lit 


v.w 


148 


INVOLUTTOK 


Algebraic  Sigfiis  of  Powers. 

170.  Since  tho  continued  proiluct  of  any  nunil)er  of  jjosi- 
tivo  factors  is  positive,  all  the  powers  of  u  positive  (piautiiy  are 
positive. 

By  §  20,  tlie  ])rodnct  of  an  odd  number  of  neirutivc  fac- 
tors is  negative,  and  the  product  of  an  even  number  is  i)ositivc. 
Hence, 

Theorem.  Tlie  evon  powers  of  iiogativo  quantities 
are  x^oaitive,  and  the  odd  powers  are  negative. 

EXAMPLES. 
(—  ((f  z=  a2  .       (_  ^,)3  _    _  f^3.       (_  (^Y  —  (ji^      e^^c. 

EXERCISES. 


Find  tlie  value  of 

I.     (-2)2.  2.  (-3)3. 

4.     (-^>)'^.  5-  (-5)3. 

7.     {—a  —  h)\  8.  {—wn)\ 

lo.     {-af'K  II.  l-hY'^'K 

13.       (—  1)2«.  14.  (_l)2nHl, 


3-  4*. 

6.  {-ly. 

9.  {  —  vif. 

12.  {-a  —  hyri-\ 

15.  (-1)2-'. 


Case  III.  Involution  of  Binomials — the  lilno' 
nilal  Theoi'cni. 

171.  It  is  required  to  find  the  n^^  pmvcr  of  a  hinomial. 

1.  Let  a  -\-  h  be  the  binomial ;  its  w'*  power  may  be  written 

{a  +  by. 

Let  us  now  transform  this  expression  by  dividing  it  by  n^, 
and  then  multi})lying  it  by  «",  which  will  reduce  it  to  its  orig- 
inal value.     AVe  have  (§  107), 

«»  \  ~a    I        \    '^  (J  ' 

Multiplying  this  last  exjnvssion  by  ««,  by  writing  this 
power  outside  the  parent liescs,  it  l)ecomes 


«"0+:)" 


(I) 


; 

I 


I 


ns 
of 


I 


INVOLUTION. 


149 


\llW' 


by  a"^, 
Is  orig- 


ig  this 


(1) 


wliicli  is  equal  to  {a  +  ly.     Next  let  us  put  for  shortness  x  to 
represent  - ,  when  the  expression  will  become 

(rt  +  hY  =  (C  (1  +  ar)«.  (2) 

2.  Now  let  as  form  tlie  successive  ])owers  of  (1  +  a:)".  Wo 
multiply  according  to  the  method  of  §  T'J: 

{1  +  xy  =  1  -{-  X 
Multiplier,  1  -f  ^ 

4-  a:    -f-  r/;2 
(1  +  xY  =z  1  +  •.>./;  +  x-2 
Multiplier,  1  -\-  x 

1  -^2x  +  x^ 

X  -\-2xi  -\-a^ 

(1  +  .r)3  =  1  +  '3x  -f-  :3.f2  H-  a-3 

Multiplier, 1  -f-_x 

'      1  +  ;}a;  +  3.r       a:^ 

a:  +  'i-^  +  'i'-^  +  X* 

(I  +  a:)4  =  1  +  4a;  +  (^.f'^  +  4.1-3  ^  ,^4 

It  will  be  seen  that  whenever  we  multi})ly  one  of  these 
powers  l)y  1  +  x,  the  coellicients  of  x,  x^,  etc.,  which  we  add 
to  form  the  next  higher  i)ower  are  the  same  as  those  of  the 
given  power,  only  those  in  the  lower  line  go  one  place  toAvard 
the  right.  Thus,  to  form  (1  +  xf,  we  took  the  cocflicients  of 
(1  +  xy,  and  wrote  and  added  them  thus : 

Cocf.  of  (1  +.^f,         1,    3,    3,     1. 

1,     3,     3,     1.  • 

Coef.  of  (1  +  x)\        lir'47~C~4,~T 

It  is  not  iKTcssarv  to  writo  flic  niinilirrs  iin<lor  onrli  other  to  ndd 
tliciii  ill  this  way  ;  we  liavi-  only  to  ndd  i-ach  iiiiiiihcr  to  the  one  on  tlic 
left  ill  tlio  .siuuo  lino  to  form  the  rorrcspondin^  nuinbcrof  tlic  line  Ixdow. 
T1iu8  wo  can  form  tlu?  conincicnts  of  the  succi-ssiv*'  jKHvcrs  of  ,/•  at  t<i^'lit 
as  follows:  The  first  fiiriiro  in  each  lino  is  1  ;  tho  uext  iB  the  cocllicifut 
of  X  ;  tho  third  the  coellicient  of  j*,  etc. 


J 


;  \ 


if  !' 


'.I 


.'    I    —  I-L-UH. 


150 


INVOLUTION. 


First  power,    n  =  1,    coefficients,  1,  1. 


{Second   " 

n  =  2, 

1,  2,     1. 

Tiiird     " 

n  =  13, 

1,  3,    3,     1. 

Fourth   " 

71  =  4, 

1,  4,     G,     4, 

1. 

Fifth      " 

n  =.  5, 

1,  5,  10,  10, 

5,  1. 

Sixth      " 

71  z=  G, 

1,  G,  15,  20, 

15,  G,  1. 

etc. 

etc. 

etc. 

It  is  evident  that  the  first  quantity  is  always  1,  and  that 
the  next  coeflicient  in  eacii  line,  or  the  coefficient  of  x,  is  7i. 
The  third  is  not  e^  idcnt,  but  is  really  equal  to 

as  will  bo  readily  found  by  trial;  because,  beginning  with 
n  =  3, 

The  fourth  number  on  each  line  is 

7i{)i  —  l){n  —  2) 
3-3 
Thus,  beginning  as  before  with  the  third  line,  where  w  =  3, 


3.2jJ.  4^3^^^ 

■     ~    2''3   '  2-3   ' 


10  =  - 


0-4.3 


2-3 


etc. 


(c) 


3.  These  several  quantities  are  called  Binomial  Coeffi- 
cients. In  writing  them,  we  may  multii)ly  all  the  denomi- 
nators by  the  factor  1  witiiout  changing  them,  so  that  there 
will  be  as  many  factors  in  the  denominator  as  in  the  numerator. 
The  fourth  column  of  coefficients,  or  (c),  will  then  be  written, 

3.2.1  4.3.2         5.4.3 


1.2.3'        i.2-3'        1.2-3' 


etc. 


4.  Wo  can  lind  all  the  binomial  coefficients  of  any  power 
wlien  we  know  the  value  of  n. 

Tiu'  numerator  and  denominator  of  the  second  coefficient 
will  contain  two  factors,  as  in  [b)  ;  of  the  third,  three  factors, 
as  in  {() ;  and  of  the  s'*,  s  factors,  whatever  .s  may  be. 

In  any  coefficient,  Die  first  factor  in  the  numerator  is  n, 
the  second  n  —  1,  etc.,  each  factor  being  less  by  unity  than  the 


1 


M 


INVOLUTION. 


151 


nonii- 
thcro 

'i-ator. 
•ittcn, 


Ipower 

iicicnt-, 
lictors, 

is  n, 
lin  the 


i'4 


preceding  one,  until  wc  come  to  the  6-'*  or  last,  which  will  he 
w  —  6-  +  !• 

Such  a  product  is  written, 

n  (n  —  1)  {h  —  2)  .  ,  .  .  {n  —  s  +  1). 

The  dots  stand  for  any  luimhcr  of  omitted  factors,  because 
s  may  be  any  number.  We  have  written  4  of  tlie  s  factors,  so 
that  s  —  4  are  left  to  be  represented  by  the  dots. 

The  denominator  of  the  fraction  is  the  product  of  tlie  s 

factors, 

1-2. .3....  5, 

each  factor  being  greater  l)y  1  than  the  preceding  one,  and  tlie 
dots  standing  for  any  number  of  omitted  factors,  according  to 
the  value  of  i\     Thus,  tlie  *'*  coefficient  in  the  ;<"'  line  will  be 

n  {n  —  1)  (m  —  2)  ...  .  (//.  —  .s'  +  1) 
~      1-2-3.....S' 


(d) 


If  s  is  gi'eater  than  |w,  the  last  factors  will  cancel  some  of 
the  i)receding  ones,  so  that  as  5  increases  from  ^i  to  «,  the 
values  of  the  preceding  coefficients  will  be  repeated  in  the 
reverse  order.  Thus,  suppose  n  =  G.  Then,  by  cancelling 
common  factors, 

C'5»4'3        G'5        ^  ^ 


1 

.2 

•  3 

.4 

0 

5- 

4- 

3. 

2 

1 

•2 

3 

4 

5 

0 

•  5 

•  4 

3 

2 

1 

1.2-3-4.5-0 


=  1. 


If  we  should  add  one  more  factor  to  the  numerator,  it 
would  be  0,  and  the  whole  coefficient  would  be  0. 

The  conclusion  we  have  reached  is  embodied  in  the  follow- 
ing equation,  which  should  be  iterfectly  nu'nu)rized  : 


(1  +  .^•)"  =  1  +  7ix  -\- 


11  ()i  —  1) 
1-2 


x^  + 


11 


(n  —  l){n—2) 


1-2.3 


+ 


11 


(n  —  l)(n  —  2)0i  —3) 


1.2-3.4 


X*  + 


•   •    •    • 


+  ^". 


m 


152 


INVOLUTION. 


f 


I 


EXERCISES. 

1.  Compute  from  tlic  formula  {d)  Jill  tlic  biuomial  cocffi- 
cionLs  for  n  =  (5,  und  from  tlieui  express  the  development  of 

2.  Do  the  same  thing  for  n  =  8,  and  for  n  =  10. 

112,  To  find  the  development  of  {a  ■-  hY,  we  replace  x 
by  -,  and  then  multiply  each  term  by  a^. 

it 

[Sec  c(piations  (1)  and  (2).]     We  thus  have 

(a  +  hY  —  a»  +  niO'-^h  +  ---{^^a^'-W  +  etc.  to  i». 

The  terms  of  tlie  development  are  subject  to  the  following 
rules : 

I.  Tlic  exponents  of  b,  or  the  second  term  of  the  hlno- 
i)ii((l,  ((re  {),  1,  2,  etc.,  to  ii, 

IJccauso  h"  is  simply  t,  d"  is  the  same  as  a"6". 

II.  T/(c  s((in.  of  the  e,vf)otients  of  ti  (did  0  is  ii  in  e((ch 
term.     Hence  the  e.vf)onciits  of  a  ((re 

n,     n  —  1,     n  —  2,    etc.,  to    0. 

III.  77/ r  coefficient  of  the  first  term  is  njiitj/,  and  of 
the  .second  if,  the  indc.v  of  the  jmirer.  J'J((eh  foHo(rii('J 
corfjieient  nitaj  1)e  foniid  from  the  next  prccedin<^  one  tjy 
mnlti plying  by  tlic  successive  fdctors, 

n  —  1  n  —  2         71  —  3 

— -  — ,    etc. 


2 


3 


IV.  Tf  h  or  a  is  negntire,  the  sign  of  its  odd  poirers 
ii'ill  l)e  cli((i}ged,  Jnit  thnt  of  its  even  poivers  will  rcmuin 
the  s((uie. 

(Compare  §  170.)     lleiico, 

{a  —  hY  =  a"'  —  iKC^-^h  +  -  -v  J- ~"-l  ^/n-2^2  _  ,_.tc., 

tlic  terms  being  alternately  positive  and  negative. 


1 


INVOLUTION. 


153 


^ 


E  X  E  R  C  I  S  E  S-Continued. 

3.  Compute  all  the  terms  of  {d  +  bY,  ii^^ing  the  hliiomiul 
eoellick'iils. 

4.  Wluit   is   the   coeilicicnt   of  U^  in  the  development  of 
{((  -H  f'Y'' 

5.  Wliiit  are  the  lirst  I'our  terms  in  the  development  of 
{•lam  +  '.Inf. 

6.  What  are  the  first  three  terms  in  the  development  of 

(1  4-  '1   ?     Wiiat  are  the  last  two  terms? 

7.  What  are  the  lirst  three  and  the  last  three  terms  of 
{a-j-y? 

8.  What  is  the  development    ?(«+)• 

9.  "What  are  the  first  i'our  terms  in  the  development  of  the 
folhjwing  binomials: 

10.  What  are  the  sum  and  difference  of  the  two  develop- 
ments, (1  +  .r)^  and  (1  —  .i)'r 

Case  IV.    Square  of  a  J*olf/no}niaI, 

173.    1.  Square  of  any  Puhjuomial.     Let 
a-\-h-\-c-{-il  +  etc., 
he  any  polynomial.     AVe  may  form  its  square  thus: 

a  I-  h  -]-  r  ^  (I  -\-  ete. 

41  -\-  h  +  c  4-  d  +  ot^c^^ 

iC-  +  ab  4-  nc  -f  rr^/  -f  etc. 

ab  +  ^'-  4-  />'■  4-  /''"'  +  etc. 

ac  4-  be  4-  t'*  4-  cd  +  etc. 

rt-f/  +  bd  -^  cd  4-  ^^^  +  etc. 

d^  +  /r  4  r-  4-  d^  +  etc. 
+  •^*(//'  4  -lac  4  -.V/^/  +  elc. 

4  'ibc  4-  2^^/  4  ete.  -I-  ^cd  4-  etc. 


■'II 


il 


154 


INVOLUTION. 


We  thus  reach  the  followin;^  conclusion  : 

Theorem.  The  square  of  a  polynomial  is  I'qual  to 
tli(^  siun  of  tlie  squares  of  all  its  terms  i)lus  twice  the 
pro(lu(;t  of  eveiy  two  terms. 

2.  Square  of  an  Entire  Function.  Smietimcs  we  wish  to 
arrange  tlie  polynomial  aTvl  its  Sf]uare  as  au  entire  function  of 
some  quantity,  for  example,  oi  x. 

Let  us  form  the  square  of  a  +  hx  •\-  cx^  -\-  dji^  +  etc. 

a  +  lx  -^  cx^  +  d.i^  +  etc. 
a  -{-  hx  -{■  cx^  +  (Ix^  -\-  etc. 


(fi  +  al)X  +  arx"^  -f  aiLc^  +  etc. 

abx  +  V\r?  +  hcx^  +  bilc^  +  cic 

acx^  4-  Icx^  4-  c^jH  -f  etc. 

(tdx^  +  J//r»  +  etc. 

«2  +  -Zahx  +  {:iac  +  <5'') ./"  +  {'i".d  +  'i^c)  .^-^  +  etc. 
We  sec  that : 


The 

coeflicient  of 

.r2 

is 

f/r  J^  lb  4-  ca. 

u 

it 

a 

2-3 

is 

ad  -\-  be  ■\-  ih  +  da. 

tt 

u 

etc. 

a 

x-^ 

is 

ae  -f  Id  +  €€■{■  dh  + 
etc. 

ea. 

The  law  of  the  products  ne,  bd.  re.  etc.,  is  that  the  first 
factor  of  each  jiroduct  is  eoniposeil  successively  of  all  the  co- 
ellitionts  in  regular  order  U])  to  that  of  the  i>ower  of  a:  to  which 
the  coenicient  l)t'longs,  while  the  second  factor  is  compotJed 
successively  of  the  same  coeilicients  in  reverse  order. 

EXERCISES. 

Form  the  squares  of  , 

I.  1  +  2.'-  +  •'^•'•^.  2.     1  +  '^.r  +  ;3rJ  -f  4.6-1 

3.  1  +  2x  +  ;if«  +  4.r3  4-  5./-. 

4.  1  +  2x  4-  ;3:/'^  +  Ax^  +  o:c^  -f  r.j^. 

5.  1  —  2x  +  3.i-'^  —  4.r\  6.     u  —b  -\-c  —  d. 

7.     3rt  +  SZ*  —  c  +  </.  o  1       7       1 

8.     a  + 0  —  J' 

a  b 


' 


i 


EVOLUTION. 


im 


>. 


» 


t :, 
I 


CHAPTER     II. 

EVOLUTION    AND    FRACTIONAL    EXPONENTS. 

174.  Def.  The  ii*^'  Root  of  a  quantity  q  is  such  a 
noDiber  as,  being  raised  to  the  n^^  power,  will  produce  q.  * 

When  n  =  2,  the  root  is  called  the  Square  Root. 

When  n  =  3,  the  root  is  called  the  Cube  Root. 

Examples.    3  is  the  4th  root  of  81,  because 

3-3.3.3  =  34  =  81. 
Ai?  the  Btiident  ah'ciidy  knows,  we  use  tlic  notation, 

9i'''  root  of  q  =  ^'q. 

There  is  another  way  of  expressing  roots  which  we  shall 
now  descrilje. 

175.  Division  of  Exponents.  Let  ns  extract  the  square 
root  of  rr*.  We  must  (iud  such  a  quantity  as,  being  niuhipliod 
bv  itself,  will  })rodtU'C  rt".  It  is  evident  that  the  required  quan- 
tity is,  a^,  because,  by  the  rule  for  mulLipUcation  (§§  GO,  IGG), 

a^  X  a^  =  «". 

n 

The  square  root  of  rt"  will  be  rt-,  because 

n  n  n     n 

a'^  X  a^  =  a^  '  ^  =  a^. 

n 

In  the  same  way,  the  cube  root  of  r<"  is  a^,  because 

II.  n  )i 

0^  X  rt'^"  X  (i^  =  a"'. 
The  fullowincf  theorem  will  now  be  evident: 

77/eorem.  The  square  root  of  a  power  may  be  ex- 
presR^d  by  dividing  its  exponent  by  2,  tlie  cube  root  by 
diriding  it  by  3,  and  tli<i  n^''^  root  by  dividing  it  by  71. 

170.  Fractional  Exponents.  Considering  only  the  origi- 
nal  definition  of  exponents,  such  an  expression  as  rt=f  would 


^i'l 


156 


EVOJMriON. 


have  no  mcaninfj,  hccanso  wo  cannot  write  a  \\  times.  But 
by  what  lui.s  just  been  said,  we  seo  that  d^  may  be  iuteri)reted 
to  mean  the  s'^uare  root  of  «^,  beeauso 

.1  n  „ 

a-  X  rts  =  a\ 
Tlonco, 

A  fnictlonal  oxpoiiont  indicates  tli(3  oxtraction  of  a 
root.  11'  the  dciioniinator  is  2,  a  ^square  rout  is  indi- 
cated ;  if  3,  a  cube  root ;  if  ii,  an  vi"*  root. 

A  fractional  exponent  lias  tliereforc  the  same  meaning  as 
tlio  radical  sign  Vj  nud  may  be  useil  in  place  of  it. 

E  ;"  E  RC  I  S  E  S. 

Express  the  folio  .  iiig  r.-^^  s  by  exponents  only  : 

I.     \^m.  2.     V{)n  +  '0-  3-     '^{^^  +  W' 


6.     \^a\ 

9.     v'(«  +  ^)"'. 


177.  Since  the  even  ])owers  of  negative  quantities 
are  positive,  it  follows  that  an  even  root  of  a  positive 
quantity  may  be  either  positive  or  negative. 

This  is  expressed  by  the  doul)le  sign  ±. 

EXERCISES. 

Express  the  square  roots  and  also  the  cube  roots  and  the 
w'*  roots  of  the  following: 

I.     {a  ■\-  hf.  2.     [a  -f  h)\  3.     a  +  h- 

4.     (•'•  +  //)^.  5.     (-i'-f//)^'-  6.     {x  +  !i)K 

lis.  If  the  quantity  of  which  tlu^  root  is  to  be  ex- 
tracted is  a  ])i'oduct  of  sjncral  factors,  we  extract  tiu^ 
root  of  each  factor,  and  take  the  product  of  these  roots. 

Example.     The  n'^^  root  of  am'^p  is  n^m'^p^\  because 
(ahN^'p'')"  =:  ((m%  hy  §§  108  and  17G. 

If  the  quantity  is  a  fraction,  we  extract  the  root  of 
both  members. 


EVOLUTION. 


1C7 


Proof. 


©"=:■ " 


§§107,108.) 


n  ,  ft 

IVciiusc  -7  takcMi  )i,  times  as  u  factor  makes  ,  ,  therefore, 

a 


by  (iL'liiiilion,  it  is  the  ii'^  root  of   .• 

EXERCISES, 

ExiHvss  the  f-quare  roots  of 


I ,       l.t  . 

tf. 

49//i 

Express  the  cube 

roots  of 

4.     27-04. 

5- 

27^3, 

7.     aPc\l\ 

8. 

8r/'« 

Ex})ress  the  n^^  roots  of 

9-     7. 

lO. 

4.7. 

12.       -            • 

bnijj"' 

1 3- 

0««^2ra. 

,jm.il,.n^-2 

IS-         ■ -.    • 

«'«« ^//l 

6.     G4.27<<^<^''. 


II.     4.7.10. 


14 


Ga^^ 


m 


c"Hln 


16.  35«  a-^«  (f«  4-  ^')^"  (.^'  —  yY  4"  (i  —  c  -f-  </)-■*«. 
Reduce  to  exponential  cxiiressions: 

17.  \a\b'^rf\  18.     ^a¥^. 

19.     ^/'iO^b^. 


11  /  ^< 


20. 


((<  —  by 

Powers  of  Expressions  with  Fractional  Expo- 
nents. 

171).  TItOorem.    Thc^  7/''  power  of  the   id'^  root  is 
equal  to  the  n*'''  root  of  the  p^'^  power. 


m 


158 


FRACTION  A  L   EXPONE^iTH. 


Ill  algebraic  langua^'O, 

or  W'T  =  {u'^)K 

Example.  (^ysf  =  '^2  =  4, 

\/H'i  =  ^/i'A  =  4; 

or,  ill  words,  llio  s(juarc  of  the  cul)c  root  of  8  (tliat  is,  the 
B(jtiaro  of  2)  is  the  cuhu  root  of  the  S(iiuire  of  S  (tiuit  is,  of  O-l). 

Cicnerul  J'roof.     Lot  us  i)ut  x  =  tiic  n^^  root  of  ^/,  so  that 

x'^  =  a.  (1) 

The  ;/*  power  of  lliis  root  x  will  then  bo  xP.  (:2) 

liaising  loth  sides  of  the  C([Uation  (1)  to  tlie  ^A  power,  we 

have 

x"P  =  aP  =  pff^  jwwcr  of  a. 

The  ;/'*  root  of  the  first  member  is  found  by  dividing  the 
exponent  by  )i,  which  gives 

?i'^*  root  of  7>'''  power  =  xP, 

the  same  expression  (2)  just  found  for  the  ;/*  power  of  the 
?i'*  root. 

This  tlioorem  leads  to  the  following  conclusion: 

1.  TIk;  exi)resyion  p 

a" 

1 

may  ni(\an  indiffiM-oiitly  the  p^'^  i)owor  of  a",  or  the  nih 
root  o{'  (fP,  those  (Quantities  bciii.ii;  identinil. 

2.  Tiu'  i)Owors  of  expressions  liavinii;  fractional  ex- 
ponents may  be  formed  by  multiidying  the  ex^HJuents 
by  the  index  of  the  power. 


EXERCISES, 


]'!xprcss  the  squares,  the  cubes,  and  the  n^^  powers  of  tho 
following  expressions : 


I.     uK 
4.     aK 


I 


;.     abK 


3.     a\ 
6.     ab^  c-'. 


IliliA  TIONA L    I'JXPniltiSIONH 


150 


ni  tn 
7.      (I'/rK 


8.     (I'iO  /'. 


tn 


9.     {a  4-  />)"  (^/.  —  /y)-».  10.     ^/  "Z*". 


I   .  I 


II.    (('*//'. 


12. 


(.'•  +  If)" 


{x-y)   ^ 

lic'diic'o  to  simple  iirodiictti  and  friictions: 

.t;"y  '7  .  14.     (<i^b^c  ")'♦. 

15.     (^r7/')  '''.  16.     \a   ")    'I. 


17 


^  e :)  • 


18. 


■♦*»■ 


/ith 

t'llts 


Uio 


CHAPTER    III. 

REDUCTION    OF    IRRATIONAL    EXPRESSIONS. 


I)i'fiiiiti(>iis(. 


ISO.  Bif.  A  Rational  Expression  is  rmo  in  wliicli 
tbo  t^ymbols  urc  only  nddcd,  snbtiafted,  nuilti|>lit'd,  or 
divided. 

All  the  opcriitions  we  have  liithortn  consldiTed,  except  the  extraction 
of  rodts,  have  Km!  to  nitional  exl)re^!siollH. 

Dif,  An  expression  wliiidi  involves  the  extraction 
of  a  root  is  called  Irrational. 

Example.     Irnitional  expressions  arc 
or,  iii  the  lunguii^^e  urex[)oiietits, 

In  order  that  expressions  may  be  really  irrational. 


\i 


100 


Iltn.  \  TIONA  L    EXPliESHlONS, 


tlicy  must  1)0  Irreducible,  tliat  is,  iucjipablc  of  being 
cxprrsscd  without  the  nidiciil  sign. 

KxAMi'Li:.     Tlio  expressions 

jirc  not    iiropcrly  in;itinn:il,  liccaiisi-  tlu-y  are  equal  to  n  y  b 
and  (i  respertivelv,  wliicli  are  rational. 

7V/'.  A  Surd  is  the  root  which  enters  into  an 
irralional  e\])ression. 

ExAMi'Li:.  The  expression  a  +  h's/x  is  irrational,  and  the 
surd  is  Vr. 

Ih'f.  IiTational  terms  are  Similar  wlien  tliey  con- 
tain tJ!'*  same  surds. 

Kx.\mi'1-j:s.  The  terms  \/'M\  7\/;J0,  (.,•  +  //)  \/;U),  are 
t^iniiiar,  beeuusu  Lhe  (pumtit)'  under  the  riulicul  .<iyu  is  oO  in 
each. 

The  terms  {^a  +  h)  Vx  -f  ^,  3\/a;  +  y,  mVx  +  y  arc 
similar. 

AsrgTojfMtioii  of  S'Miilar  Terms. 

181.  Trratioiud  terms  may  he  a'r;j:reguted  by  the  rules  of 
§§  o4-r)0,  the  surds  being  treated  as  if  they  were  single  sym- 
bols.    Hence: 

IJVirn,  similar  iiTdHonnl  frrins  nra  cnnncctrd  hy  fliG 
sij^tis  -\-  or  —,  tJic  corfJiricittH  of  thr.  sii)iil((r  sitrds  indij 
l)C  (tddi'il,  (Hid  lhe  surd  Hsrlf  affixed  to  their  sum. 

ExA.Mi'Li;.     The  sum 

« V(.''  -f  //)  -  /V(.''  +  //)  +  oV{.r  +  y) 
may  be  transformed  into  {a  —  h  -{■  W)  ^/{x  -\-  y). 

EXERCISES. 

Reduce  the  following  expressions  to  the  smallest  number 
of  terms: 


lUn. I TTONA L    KXPRKSsmXSt. 


101 


an 


are 


2.     {W{x  ^  y)  +  ay/{x  -  ^)  +  -^  («  +  h)  \/{x  -f  //) 

-.*i(«  -\-b)^{x-}i). 

4.  ("  +  <''')  V-^  +  {(t  — 1>)  V^h 

5.  \U  {a  —  b)  +  {/j  —  (•)  V^  4-  (c  —  a)  Vx, 

6.  «\/^-  —  \^x  4-  '^rt  \/.<;  —  (r«  4-  ^)  \/.«^. 

7.  '  Va;  — a\/^  +  GVu- —  fV^  +  .,  V^' 

4  J 

„      a  -{•  0    /        „     /        n  ■\-  h    /  / 

8.  —  ,  -  \x  —  Gcv-f  —  -  ..  -  yx  +  \x. 

9.  V^  —  V^  +  ('«  —  b)  V -i'  +  — -., ■  V.^'. 

/  ,    /  /         n(rr  — /;)    /         1     / 

10.  Vrt  —  h\a—  yx  -\ — ^~—.  —  ya  —  .,  ya. 

11.  Wx^yx  -^  -^ — -' yx, 

1 2.     4  V^  —     \/ /'  +  (r/  —  b)  Vx. 
o 

Factoring:  Surds. 

J83.  Irrationiil  expressions  niiiv sometimes  he  t ran ^^ formed 
so  as  to  have  diirereiit  ex])ressions  lUKk'r  the  radical  sign,  hy 
the  inelhod  of  g  liS,  aiti)lyiiig  tlie  foUowiiiL?  theorem: 

Theorem.  A  root  of  the  product  of  several  factors 
is  equal  to  the  ])ro(luct  of  their  roots. 

In  the  langiu^cfe  of  Algehra, 

y^dbcd,  etc.  =  y^a  Vb  \/c  y  d,  etc. 
=  fO'h"  Ad'\  etc. 

Proof.  By  raising  tlie  memhers  of  tliis  equation  to  tlie 
n^^  power,  we  shall  get  the  same  result,  namely, 

a  X  b  X  c  X  d,  etc. 

Example,    a/30  =  \/o  Vs. 
11 


m 


102 


IliliA  TIONAL    EXPEESSIOXS. 


EXiZRCISES 

Prove  tlie  following  c'f|Uutioii.s  by  conipnting  both  sides: 
Vroof.     V-i  V49  :=  2-7  =  14,  and  VlOG  -  14. 

Vi  Vi)  —  V;)n. 

a/4  V-'y  ==  \/4-25. 

a/-'5  V'jO  =  V^^o-aO. 
Express  with  a  single  surd  the  products: 
I.     ^/{a-\-h)v\a  —  b). 


bOLUTiox.     V{a  +  I)  V{a  —  h)  =  V{it  +  b)  {a 

=  Via-'  -  b'). 
2.      VT  Va.  3 

4.     V^/  a/('^  +  >j).  5 

6:    V(r  +  1)  \/(.'- -  1). 
7.    V(/'  +  1)  V(.r+  1)  V(.f 

8.  [{a  +  b)U"  —  b)y\\ 

9.  Ui-2+i)^^_^.i)^.(.,_i)?,f. 


"^) 


v'7  Va. 

Va  Vb  V{(t  +  b). 
1). 


1S;{.  If  wo  can  S('i)anito  tl'o  quantity  nndcr  the 
radical  si<2:u  into  two  factors,  one  of  which  is  a  perfect 
square,  we  may  extract  its  root  and  affix  the  suid  root 
of  the  remaining  factor  to  it. 

EXAMPLES. 

'vA/Vy  =  Vtr-  Vb  —  nVb. 
Vffb  V'f'  =  Vc'br  =  aVbr. 

V\-i  V('>  -  V'rl  =  a/:3<;  V'i  =  oV^. 
V{-^((^  +  Srt^Zi  —  HUt^)  =  V-iff^ifT+Jb^-lac) 

:=  '^nVi'f  +  'ib  —  ^ar). 


lik 
ro(j 


bii 


' 


re; 


ILiEA  TIONA  L    EXPUEfiSlONS. 


1G3 


Icsi 


EXERCISES. 

Reduce,  ■svlieii  i)ossil)le  : 


tlie 
ri(!ct 
root 


I. 

3- 

5- 

7- 

9- 
II. 

13- 


a/8. 

\/l*-'8. 

\/'/<^  V'^v(  V^bc. 

a/4  a/7^. 
a/1T5. 


2.  a/3-2. 

4.  V :{  V-7. 

6.  a/;>  A/:;i. 

8.  A/(r  +  1)  \^{x  +  1). 

10.  Vl  •"><>. 

12.  V^6-^((<  +  ^). 


V  108. 

A/(»:-.f  +  Srt^'.r  +  b\c). 
Hero  the  luantity  luuler  the  radical  sign  is  equal  to 
{((■  +  'idb  +  l)^)x  r=  (a  +  fc)-  .r. 

In  quostionsof  this  class,  the  iM-ginncr  is  apt  todivido  an  expression 
like  y/n  +  6  +  c  into  y/a  -\-  '\/b  +  'sjc,  which  is  wrong.  Tin-  Hiiuuro 
root  of  the  sum  of  several  (luantities  cannot  be  reduced  in  this  way. 

14.     a/«-'//  4-  4f?//  4-  -1//.  15.     's/^mh  -f  '6mz  -\-  \z, 

Ift'duco  and  add  tliu  following  surds: 

16.     'lA/-i-0A/H4-10\/3.'->.     17.     \/l-2  4- \/'-2r  +  a/75. 
18.     ^fXa  —  'Xy/u.  19.     l-^.j^  —  4')'' —  80^. 

20.     -\/8l-vli)^.  21.     (rt^Z^)»  4- (r(V')i. 

3Iii1(ii>li('ati<>ii  of  Irrational  Expressions. 

18 1-.  Irrational  polynomials  may  Ik-  mnltiplii'd   by  com- 
biniiii^  tlio  for' going  j)riiK'ip1t\s  with  the  rule  of  §  18. 
The  following  are  the  forms  : 

To  multii)ly  a  4-  h\/ x  by  m  4-  n\'  if. 

(I  {»i  4-  iiVj/)  ■=  <nii  X  (tw^ If. 
hVx{m  4-  ws/ u)  =z  hmVx  4-  hnV.11/. 

The  produeL  i.s  uin  4-  an\^ ij  4-  bniy/x  -\-  bu\' xij. 

EXERCISES. 

Perform  (he  f(dlowing  mnltiidieation.s  and  reduce  liio 
results  to  the  simplest  lorm  (compart'  i^  S(i): 

I.     (3  4-  3A/r))  (.•>  -  3V^>).         2.     (;4-3\/3-.0  (0  -  'o's/'X). 


!*; 


1G4 


lliEA  riONAL    EXPRESSIONS. 


3.     {a  +  ^/b)  {a  —  Vl>).        4-     ( V^  +  VA  +  V<-  +  Vdf. 
7.     0«  +  «"-0'.  8.     (^ri -«'•)*• 

9.    [«  +  ^'V(^-  +  2/)][r?-i\/'(.c  +  y)]. 

10.  [//t  -I-  ?i\/(rt  -I-  /;)]  [///  —  ny/{a  —  />)]. 

11.  [x  +  a/(.':2  -  1)J  [x  -  ^/{x^  -  1)]. 

12.  [(^J4-])i4-/,J[(ia  +  l)i_Z,]. 

Expressions  may  often   be  transformed  and  factored   by 
coni!)ining  tl»e  foregoing  processes. 

Example.     To  factor  nx^  +  hx^  +  cx^  4-  dx^,  we  notice 

*^^'^^  a;5  =  a:ii:3,         2-5  =  x-^x%     etc. 

so  tliat  the  expression  may  be  written, 

(U^x-=>  +  bx:'x^  -\-  cxx^  4-  dx-^  =  {((x^  +  hx^  +  ct  -f  </)  x^, 

EXERCISES. 

lieduce  tlio  following  expressions  to  prodncts: 


13.  2  +  a/2. 
15.  {a  +  f>)l 
17.     X  —  1/  —  Vx 


14. 
16. 


:}-  4-2.;3i. 


V//  4-  <if  —  bif. 


Ivedncc  to  the  lowest  terms : 

2  ^/a  A-~b 

—    .  10. 

y2 
a 


1 8. 


19. 


a  +  b 


21. 


rA 


a;  4-  Vrt  —  « 

.r  —  \^a  —  X' 


20. 


22. 


ax^  4-  i^^ 


a  -{-  b    ' 


1S.">.  Riifionalizhifj  Fracfio)h9.  The  quoti(>iit  of 
two  surds  iiiav  be  c'X])r<'ss('(l  jisa  fnictioii  with  ai'.'itioiial 
imuK'nitor  <»i'  ji  rational  (Iciioniiiiator,  hy  imiltiplyiiig 
both  tonus  by  the  X)i'op('r  iiiulti])lit'r. 


ExAMTLE.     Consider  the  fraction 


v? 


\\ 


IRRA  TIONA  L    EXPRESSTOKS. 


165 


^Multiplying  l)oth   terms   by   y/'H,    the    fraction    becomes 

— ~,  uml  lias  the  rational  denominator  7. 

'A 

^Fultiplving  bv  ^J'o,  it  becomes     '     ,  and  has  the  rational 

numerator  5.  ^  '^'^ 

The  iiuniorator  or  denominator  may  also  he  made 
rational  when  tliev  both  consist  of  two  terms,  one  or 
both  of  which  are  irrational. 

Let  us  have  a  fraction  of  the  form 

A  +  D\/B 

in  which  the  letters  A,  D,  /',  (>,  and  I!  stand  for  any  algebraic 
or  numerical  expressions  whatever.  If  we  multiply  both  nu- 
merator and  denominator  by  J*  —  QV-lh  the  denominator 
will  become 

The  numerator  will  become 

AP  +  PDVB  -  A  qVu  -  dqVbr. 

so  that  the  value  of  the  fraction  is 

A  r  -f-  /yV/>'  -  A  qVr  -  DQV/in 

pi -QUI 

EXERCISES. 


Ikcilucc  tiu'  ."jllowing  fractious  to  others  having  rat iotud 
denominators: 


I. 


ft  -\-  VI 

Vb 


2.       -: 


5- 

8. 


<i 


2\/l8^ 

n  —  \/x 
a  -f  Vx 


3- 
6. 


V-r  +  Vy 

Vx  —  Vtj 


100 


TEJIFEVT  SQUARES. 


lo. 


a  -f   \\x  +  y) 


a- 


II, 


13. 


15- 


Vs  — V3 


■\/x  +  a  +  Va'  —  a 
"s/x  '\-  a  —  Vx'  —  If' 


Perfect  S(iiiJiros. 

1S(».  Dcf.  A  Perfect  Square  is  an  expression  of 
which  tiic  s(iiiar(»  root  can  be  fornuMl  without  any  surds, 
oxci'i)t  such  as  are  already  found  in  the  expression. 

ExAMPLKS.  4wrS  4r<2  -f  \a  -\-  1  are  peneci  S((Uju'o.s,  bo- 
caiiso  llu'ir  sciuiire  roots  are  'im\  'Za  -\-  1,  exprcsisious  without 
the  ni(li(!ul  sign. 

Tho  expivsision  a  +  "l^/ub  ■\-  b,  of  wlilch  tlic  root  is 

V^'  4-  V/^ 
may  also  ho  r('<,'ar(U'(l  as  :i  perfect  .<(puire,  Ix'causc  tlio  surds 
^/a  4-  \^b  arc  in  the  prothict  'l^/ab. 

Cnterion  of  a  Perfect  Square.  'V\\v  (|iiestion  wlietl^'^r  a 
trinomial  is  a  pt'i'fcct  s(juare  can  ahvays  be  decidetl  ])y  ci'iiipar- 
iiig  it  witii  tiie  forms  ol'  j?  80,  namely : 

«a  4.  ^Zab  +  U^  =  {a  +  by, 
Ot  «2  _  >y,,i,  4-  ^2  —  (rt  _  b)\ 

Wc  see  that  to  be  a  peii'ect  squa;-r  ••  trinomial  must  fulfil 
the  following  conditions: 

^.)  Two  of  its  three  U^rms  nuist  be  ])erfect  scpiares. 
^       (2.)  The  reniaiuing-  term  must  be  (Hpuil  to  twice  the 
produ<*t  of  tlie  sijnare  roots  of  the  other  two  terms. 

WIk'II  these  conditicms  are  fulfilled,  the  scpiare  root 
of  thr  trinomial  will  1k'  the  sum  or  dilFerence  of  the 
S(piafe  root^  of  the  terms,  according-  as  the  i)roduct  is 
])ositiv'c  or  negative. 

T\w  root  may  have  either  sign,  because  the  scpuires  of  posi- 
tive ai*»l  "K'g.itive  (puuitities  have  the  same  sign. 


the 


COyiPLF.TTNG    THE  SQUARE. 


1(57 


npur- 


fuini 

•the 

root 
>r  tlio 
net  is 

f  |insi- 


If  liie  torma  which  are  perfect  squares  arc  hofh  iKgative, 
iiic  trinomial  will  he  the  iK\i,'ativt'  of  a  perfect  S([!iure. 

EXAMl^LCS. 

^/Ti^+llub  -f  l'~  —  a  +  b  «)r  —  {<i  -f  b). 
^/ifi  —  'Zab  -{■  0^  —  a  —  b  uv  b  —  a. 
_  ci^  +  '^ab  —  Ui=^—{a-bf-—        {b 

EXERCISES. 


ay. 


Find  "which  of  the  follo\viii<jr  exiircssions  arc  perfect  squares, 
and  extract  their  s(piarc  roots: 


I.     i)  +  12  +  4. 


3- 
5- 

7- 

9- 
1 1. 

15. 


Ax*  -f-  ^-^^  +  ,  • 
•t 

4«2«  -I-  V2a"b"  4-  0//^«. 

•± 
7)1  +  2tn^n^  -I-  ??. 

rt  +  4r/i/>i  +  4i. 


2.  a-  4-  4a:  +  4. 

4.  r<2  -f  f/^  —  ^2^ 

6.  a^  4-  ^^//y  —  /A 

8.  rt2^  —  2^/^»tv/  +  c^(P. 

10.  rt'^  —  2r/.r  -f  ?/2. 

12.  rr  —  2  +  cr*. 

14.  ('>I:>n''"  +  //-  4-  ^m^". 


VXi-Y  +  'J^^  —  '^'Irijz.     16.     9m8'»  —  2/y«^'V^(/  + 


—  • 


To  Coiiiploto  tlio  Square. 

187.  If  one  tcnn  of  a  hinoniial  is  a  perfect  sqnaro, 
such  a  t<'i-iii  can  always  Ix^  added  to  the  hinoniial  that 
the  trintunial  thns  formed  -^liall  ho  a  pcTfect  square. 

'J'his  operation  is  caUed  Completing  the  Square. 

Proof.  Call  a  the  root  of  the  term  which  Ik  a  perfect 
.«(|iiare,  which  term  we  su])posr>  the  //rs7.  and  call  in  the  other 
term,  .so  that  the  given  hinomial  shall  I>o 

a'  +  )n. 


)n- 


Add  to  this  hinomial  the  trrm    .   ,,  and  it  will  become 

•la- 


«2  4-  "'  + 


7ir 

4a^ 


108 


COMPLETING    2UE  SQUARE. 


\i\ 


|i'; 


I  It 


Tlii.s  is  a  perfect  .S(iuare,  luimcly,  the  square  of 


fi  + 


in  ^ 


'4a 


that  is. 


a-  +  m  -f 


=  {'■  ^  £/ 


Hence  tlie  follow  injj 


Rule.  ,'/(hI  to  the  hinnmiul  the  Sffunre  of  the  second 
term  divided  by  four  times  the  first  term. 

Example.     Wiiat  term  must  be  added  to  the  expression 

7^^\ax  (1) 

to  mal<c  it  a  perfect  square  ? 

The  rule  gives  for  the  term  to  be  added, 

Tlicrcfore  the  required  i)erfect  s-joare  is 

x^  —  \ax  +  4rt-'  =  {x  —  2«)2 

We  may  now  transpose  ^*fi,  so  that  the  left-hand  member 
of  the  equation  shall  be  the  original  binomial  (1).     Thus, 

x^  —  \ax  =.  {x  —  2rt)'  —  4«2. 

The  ciii^inal  ])in()iniiil  is  now  exj  .vssed  as  the  difTercncc  of 
two  scinares.  Therefore,  the  aTK»ve  process  is  a  solution  of  the 
problem  •  lltirim)  a  binomial  of  irhich  one  term  is  a  perfect 
square,  to  express  it  a«  a  difference  of  tteo  squares. 

EXERCISES. 

Express   the  following  binomials  as  differences   of    two 

squ.ires: 


I.  x^  4-  '^'U' 

3.  X^  -t~  (iflf.f. 

5.  i.X^  +  ±/-v. 

7.  li\xi  ~\~  ^'iinx. 

9.  ^r.c^  +  %ih:. 

II.  m^x^  -i~  L 

<3.  ^'^,+  1. 


2.  .r*  -i-  ix}/. 

4.  4x^  -f  4.ry. 

6.  Ojt*  +  ax. 

8.  JT*  +  4.r. 

10.  Irx^  f  2. 

I ::.  iV/V  -f  bx. 

•4.  aJ:^-«'A 


IliliA  TIONA  L   FA  CTORS. 


100 


(1) 


IOC  of 

)i'  the 
nfcd 


two 


. 


Irratioiiiil  Factors. 

18S.  Wlien  we  introduce!  surilri,  many  expressions  ran  ho 
factored  wliicli  have  no  rational  factors.  The  following 
iheurem  nia}'  be  ai>i)lied  for  this  i)urpo.sc : 


Tli('  difFc 


of 


t\V( 


qnantitios  is 

equal  to  the  piHjduct  of  sum  and  difleieuce  of  their 
square  roots. 

In  the  language  of  algebra,  if  a  and  b  be  the  quantities,  wo 
shall  have 

a-h  =  (rti_i?5)(ai  +  hV), 

which  can  be  proved  b}^  multiplying  and  by  §  80,  (3). 


Factor 


EXERCISES. 


I.     m  —  n. 


3- 

5- 

7- 
9- 


am  —  bn. 
x^  —  m. 

(.r  —  aY  —  V  {m  —  n). 


2.  m  —  1. 

4.  ^ahn  —  9. 

6.  x^  —  (m  +  w). 

8.  a;2  —  (ni  —  w). 


(a  +  bf  —  (4/  —  (j).     I  o.     a;2  +  ±cy  -\-f—  {711  -\-)t)K 

Find  the  irrational  square  roots  of  the  following  expressions 
by  The  principles  of  §  18G  : 


II. 

rt  —  2  +  a-K 

Alls,  a^  —  a~^. 

12. 

X  —  'i^xy  +  y. 

13- 

4  +  i  \/3  +  3. 

14- 

9  _l_  5  _  oVd. 

15. 

4rt  i-  />  —  4«U2. 

16. 

0  +  ^^  +  2  (a  +  <5')-i 

;c4-ic2. 

17- 

3  -f  2\/l5  4-  5. 

18. 

3  +  5  —  2VI0. 

19. 

4"^4  ^      2    • 

20. 

a  —  2\/a  -f  1. 

21. 

rt  —  2a«  +  at. 

22. 

,        1 

a  -f  2a5  H-  nr- 

^Z- 

7               ai 
rt»  —  rt  -1-  --. 

4 

24- 

«  +  «  +  «. 
4  ^  3  ^  9 

»5- 

KJ  ^  4  ^  4 

26. 

^r^  -f  2  -f  ^r=. 

J 

27. 

4^  _  8  +  ^x-\ 

2S. 

a 

4 

+  b 

BOOK    VI. 

E  Q  UA  TI O  N  S   R  E  O  UIR  IN  G  IR  R .  I 
TIONAL     OPERATIONS, 


CHAPTER    I. 

EQUATIONS    WITH    TWO    TERMS    ONLY. 

180.  In  the  present  chapter  wc  consider  ('(pialions  wliieli 

contain  only  a  sinule  j)o\ver  or  root  of  the  unknown  (pianlity. 

Such  an  equation,  wlien  reilucucl  to  the  normal  form,  will 

be  of  the  form 

^.<»  +  /?  =  0. 

By  transposing  B,  dividing  by  A,  Jind  putting 


a  = 


B 

A' 


the  equation  may  be  written, 

a;»  —  a  —  0. 

or  x^  =1  a,  (1) 

Here  n  maybe  an  intcpjer,  or  it  may  represent  some  fraction. 
Such  an  c([uation  is  called  a  Binomial  Equation,  because 
the  exi)ression  u."  —  a  is  a  binomial. 

Solution  of  a  Binoiuiiil  Equation. 

190.  1.  WJien  tlic  vxpouoif  of  x  is  a  whole  nnmhcr.  If  we 
extract  tiie  ;/''  root  of  botli  members  of  the  ctiuation  (I),  ihcso 
roots  will,  by  Axiom  V,  still  be  eijual.  The  w'**  root  of  .c"  being 
Xf  and  that  of  a  being  a",  we  have 


and  the  equation  is  solved. 


X  =  a^, 


I 


lUNOMlA  L    Kq  UA  TIONS. 


171 


1. 


2.   When  the  exponent  is  fractional.    Let  the  cf|Uiitiou  be 

M 

j^  =  a. 
\{X\A\\'^  Itolli  nu'iiiliers  to  the  w'*  power,  we  luivo 

Extracting  the  wi^'*  root, 

n 

X  =  a»K 

If  tljo  niimcriitor  of  the  c.\])ouent  is  unity,  we  only  have  to 
sui)pose  ni  =  1,  which  will  give 

X  =  rt». 

II(>nro  the  hiiioniial  O(|n;ition  always  admits  of  solution  hy 
forming  powers,  extracting  routs,  or  hotli. 


(1) 


I 


Special  Forms  oi'  liiiioinial  E<iiiati(>iis. 

A/.  AVIion  tlio  exponent  n  is  an  iiitonci-,  Die  equa- 
tion is  called  a  Pure  Equation  of  the  (1(  uree  Ji. 

\Vlion  vi  =  2,  the  eciuation  is  a  Pure  Quadratic 
Equation. 

When  n  =  3,  the  equation  is  a  Pure  Cubic  Equa- 
tion. 

EXERCISES. 

rind  the  values  of  .r  in  the  following  equations: 


P 
I.     ^  =  q. 


2. 


(f_±_b 

"  r 


c. 


X 


9        x^ 

4.  =:      -. 


6. 


.r^  —  n(f 


V*  —  a 
x^        1f^ 


vrl-  b 


8.     ^  =    .. 


3- 
5- 
7. 

9- 


Ana.    X  =  - . .• 
2^ 


a 

0 

-A  -  b  ~ 

X^'  — 

a 

X  —  'ia  _ 

2.r~ 

h 

—  t 

X  —  a   ~ 

X  — 

b 

a  i-  b 

V 
X<i 

\ 

in        — — 

a  —  b 

_     h 

V->'  +  ff'^ 

a 

a  +  b  \/x—7i 


172  rOSTTIVK  AND   NhUlATIVh:   iidOTS. 

In  tho  last  examj»lo,  clt'nring  the  ociuation  of  fractions,  wo  shall  have 

^¥^^ti*  -  b'  -  a\ 
iff  (X' -  a*}^  =  i' -  a\ 

Wo  W|unro  both  HidcH  of  thin  L-quation,  which  gives  another  in  whieh 
I*  only  appearH. 

lo.     (,(•  —  ^/)'  =:  />'.  II.     (.t'J  —  rt=)5  =  mx. 

12.      (V^'  —  V^)-'   =  HU*. 

Posiilvo  and  Xojjrttlve  Roots. 

101.  Since  tho  sfpiiiro  root  of  a  <|Uiiiility  uiiiy  l>o  oitlior 

positive  or  nugulive,  it  follows  that  when  we  iiuvo  an  t'<iuation 

BiK'h  a.s 

ar-'  =  a, 

and  extract  the  Sfjuarc  root,  we  may  have  citlicr 

rr  =  -f  rts, 

m  X  —  —  a^. 

IToncp  there  are  two  roots  to  every  such  cruiation.  flio  one 
positive  and  tiie  other  negative.  We  e.\pre;^s  this  pair  of  routs 
by  writing 

X  =  ±  «^, 

tlio  expression  ±  a^  meaning  eitlier  +  a^  or  —  n^. 

It  niifi;lit  sc'cni  that  since  the  square  root  of  x-  ia  either  +  J*  or  —x,  wo 
should  write 

±  a;  =  ±  a*, 
having  the  four  equations,  x  —  <t\ 

X  =  —  a*, 

—  X  =  +  «*, 

—  X  =  —  a^. 

But  the  first  and  fourth  of  these  equations  pive  identical  values  of  .r 
by  simply  changing  the  sign,  and  so  do  the  second  and  third. 

PROBLEMS    LEADING    TO    PURE    EQUATIONS. 

T,  Find  (liroo  tiunilior?,  sucli  that  tlie  second  sliall  ho 
double  llie  first,  the  tliird  one-third  the  secuud,  and  the  .sum 
of  their  scjuares  11)0. 

2.  Tlie  sum  of  the  Sfpinrcs  of  two  numbers  is  .'ino,  and  the 
difference  of  their  scpiares  81.     What  are  tlie  numbers? 


8  11 


pnonLr:}fs. 


173 


fe) 


no 


3.  A  ]ol  of  land  rojitaiiis  1(14.")  sfjuaro  foct,  and  ifs  Icn^rth 
oxpimmI-  its  hreadlh  by  U  IVct.  W  liaL  aiv  tlic  Icii^'tli  and 
bivadlh? 

'I'd  hdIvc  tliirt  ('(iiintion  ns  a  !»inoniinl,  tnke  tlio  nx-nn  nf  the  li-n^'tli 
nnil  lirt'iiiltli  iistlic  iinkiiown  (|uaiitity,  !<(i  thut  thu  Icii^tii  hIiuU  b«*  ii.s  luucli 
grt'iili T  tlmn  tin*  iiicim  uh  tlic  lirnultli  irt  Icsa. 

4.  Find  :i  mnnborsucli  that  ifO  ho  added  to  and  subtracted 
from  it,  the  product  of  the  .sum  and  tlilTcrencc  .sluill  be  1T'>. 

5.  Find  a  number  such  that  if  rj  be  addc*!  to  it  afid  sub- 
tracted from  it  the  product  of  tlic  .sum  and  dilTerenie  aliall  bo 
'^a  +  1. 

6.  One  numl)er  i.s  double  anothrr,  and  the  dilTerencc  of 
tlieir  .s<|uares  is  Wri.     What  are  the  numbers  ? 

7.  One  number  is  8  times  an<)ther,  and  tlie  sum  of  their 
culie  roots  is  J*^*.     W  bat  are  the  nund^ers  'f 

8.  Find  two  numbers  of  ubieh  «»ne  is  .T  times  the  other, 
and  the  s(|uure  root  of  their  bum,  muinpbed  by  the  lesser,  is 
e<iual  to  I'-iS. 

9.  Wbat  two  numbers  arc  to  each  other  as  3  :  3,  and  the 
sum  of  their  s(juarcs  =  '^^2')? 

NoTK.     If  we  represent  ona  of  the  numbers  by  2t,  tlie  othiT  will  be  3.r. 

10.  What  two  numlu'rs  are  to  each  other  as  711  :  ;/,  anil 
the  .^i|iiare  of  \\\v\v  dilTerencc  equal  to  their  sum  ? 

11.  W  hat  two  numliers  are  to  each  other  as  1)  to  7.  and  tho 
cube  root  of  their  dilTerencc  multiplied  by  the  i;<piarc  root  of 
their  sum  e<|ual  to  1(5  ? 

12.  Find  X  and  1/  from  the  equations 

af'  -f-  ////2  =  r, 
(I  .i-  -\-  0  y-  =  c. 

13.  The  hy])ot]ienuse  of  a  rifj^ht-aiifrled  trianirle  is  20  feet 
in  len^'lh,  and  the  .-um  of  the  sides  is  :Jl  feet.     Find  each  side. 

NoTK.  It  is  shown  in  (Jconu'try  thut  the  S4|unrv  of  the  hypotheniiHO 
of  n  liirlit-an^U'il  trianj.rle  is  e(|UHl  to  the  Huni  of  tlie  wiuares  of  the  other 
two  sides.  In  tlif  ])rrsriit  prublriii,  take  for  thf  unknown  i|uaiitlty  tho 
aiuoiint  l»y  which  eai-h  unknown  side  ditl'tTS  from  half  their  sum. 

14.  Two  ])(>iiits  start  out  tou't'ther  from  the  vertex  of  a 
rlLdit  allude  alnnsi;  its  respective  side-;,  the  one  moving,'  )n  feet 
per  second  and  the  olher//  iril  per  second.  lliiwIolii(  will 
they  re(|uire  to  l)e  c  leel  apart? 

15.  By  the  law  of  Jailing'  bodies,  the  distance  f.illen  i'J  pro- 
])ortioiial  to  tbc  S((U,iri'  of  the  time,  and  a  bndv  falls  Hi  i'tct 
the  lirsL  jsecund.     llow  long  will  it  rcipiire  to  fall  h  feel*:' 


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174 


Q  UADllA  TIG  EQ  UA  TIONS. 


CHAPTER     II. 
QUADRATIC     EQUATION  S. 

192.  Def.  A  Quadratic  Equation  is  one  which, 
when  reduced  to  the  normal  form,  contains  the  second 
and  no  higher  power  of  the  unknown  quantity. 

A  quadratic  eciiiation  is  the  same  as  an  equation  of  tlie  second  degree. 

Def.  A  Pure  quadratic  equation  is  one  wliich  con- 
tains the  second  power  only  of  the  unknown  quantity. 

The  treatment  of  a  pure  ([uadratic  equation  is  given  in  the  preceding 
chapter. 

Def.  A  Complete  quadratic  equation  is  one  which 
contains  both  the  lirst  and  second  powers  of  the  un- 
known quantity. 

The  normiil  form  of  a  complete  quadratic  equation  is 

ax^  -\-  bx  +  c  =  Q.  (1) 

If  we  divide  this  equation  by  a,  we  obtaia 

x^j^^-x-\--  =  0.  (2) 

a         a  ^ 


Putting,  for  brevity,      -  =  p, 


'•ii 


the  equation  will  be  written  in  the  form, 

0^2  ^  px  +  q  =  0.  (3) 

Def.    The  equation 

a?'  +  px  -{-  q  —  0 

is  called  the  General  Equation  of  the  Second  Degree, 
or  the  General  Quadratic  Equation,  because  it  is  the 
form  to  which  all  such  equations  can  be  reduced. 


f 


(2) 


(3) 


q  UADllA  TIV  Eq  UA  TIONS. 


175 


Solution  of  Ji  Complete  Qujidratic  Eqiuition. 

1(),*5.  A  quadrdt'iG  eqnatinii  is  solved  hi/ adding'  such 
a  (iitanUty  to  its  tico  menibers  tJiat  the  inenihrr  con  tain - 
in(^  the  unlcnowii  qaantity  shall  he  a  perfect  s(/uare. 
(fl87.)  * 

We  first  transpose  q  in  the  general  e»|uation,  obtaining 

x^  +  px  =  —  q. 

rfi 
We  then  add  ^  to  botli  members,  making 

The  first  member  of  tlie  c(iuation  is  now  a  perfect  square. 
Extracting  the  square  roots  of  both  sides,  Ave  have 


x-\-\ 


P  _ 


'P' 


2 


-q. 


From  tliis  equation  we  obtain  a  value  of  x  wliich  may  be 
put  in  either  of  the  several  forms. 


X 


a  ^  V  4 


p   ,   Vp^  —  ^q 


X 


.A-p±Vp'-4:q)' 


If  instead  of  substituting  7)  and  q,  we  treat  the  equation  in 
the  form  (2)  precisely  as  we  have  treated  it  in  the  form  (o),  wo 
sliall  obtain  the  several  results. 


,       b  1  IP       1  Ir 

x'  +  -'  X  +        ,  =        ., 

(t  -i  ((■       4  ^/■ 


a 


and 


1    ,      //i:        r\ 
^--  ^za  ^  V  Urt^      a) 

—  — ^  ±  \/(/y^  —  4fl!c) 


176 


QUADRATIC  EQUATIONS. 


J 


?' 


1 

is 


194.  The  equation  in  the  normiil  form,  (1),  may  also  be 
solved  by  tlie  following  process,  which  is  sometimes  more  con- 
venient. Transposing  c,  tnd  multiplying  the  equation  by  a, 
we  obtain  the  result 

d^x^  4-  aT)x  =:  —  ac. 

To  make  the  first  member  a  perfect  square,  we  add  j-  to 
each  member,  giving 

a^x^  +  ahx  +  -  =  - —  ac. 
4       4 

Extracting  the  square  root  of  both  sides,  we  have 

ax-\-^  =  ^  V{i>^  —  4:ac), 

from  which  we  obtain  the  same  value  of  x  as  befo.e. 

195,  Since  the  square  root  in  the  expression  for  x  may  be 
either  positive  or  negative,  there  will  be  two  roots  to  every 
quadratic  equation,  the  one  formed  from  the  positive  and  the 
oilier  from  the  negative  surds.  If  we  distinguish  these  roots 
with  u^  and  x.^,  their  values  will  be 

-  —  ^  +  Vjh'*  —  4^c) 

'  !*  }  (4) 


x^  = 


%Va 


2a 


We  can  always  find  the  roots  of  a  given  quadratic  equation  by  sub- 
stituting the  ooofRcients  in  the  preceding  expression  for  x.  But  the  stu- 
dent is  advised  to  solve  each  sei)arate  eijuation  by  the  process  just  given, 
which  is  embodied  in  the  following  rule ; 

I.  Reduce  the  equation  to  its  normal  or  its  general 
form,  as  may  he  most  convenient. 

II.  Transpose  the  terms  which  do  not  contain  x  to  the 
second  member. 

III.  //  the  coefficient  of  x"  is  luiity,  add  one-fourth 
the  square  of  the  coefficient  of  x  to  both  members  of  the 
equation  and  extract  the  square  root, 

IV.  If  the  coefficient  of  x''^  is  not  unity,  either  divide 
Jjy  it  so  as  to  reduce  it  to  unity,  or  multiply  all  the  terms 


% 


Q  UA  DRA  TIC   EQ  VA  TIOXS. 


177 


to 


(4) 


L 


1)7/  such  a  factor  that  it  shall  hrrnme  a  perfect  square, 
and  cuDiplete  the  square  hy  the  ride  of  §  18T. 


Solve  the  equation 


EXAMPLE. 


x  —  l 


=  2x. 


x~-Z{) 

Clearing  of  fractions  and  transposing,  we  find  the  equation  to  become 
2xi  —  4:lx  +1  =  0, 

„  _  41^  ^  _  1 
2    ~       2' 
Adding  \  the  square  of  the  coefficient  of  .v  to  each  side,  we  have 
41  1081        1081        1         1073 


^'-T^  +  T(r 


and 


10  2  ' 

Extracting  the  square  root  and  reducing,  wo  find  the  values  of  x  to  bo 

:r/=^(41  +  ViGTS), 


x^  =  ^(41_^/lo:3). 


Ufiing  the  other  method,  in  order  to  avoid  fractions,  we  multiply  the 
equation  (5)  by  2,  making  the  equation, 

4a;2  —  82:e  =  -  5. 

41'      IGSl 
Adding  —  —  —.-  to  each  side  of  the  equation,  we  have 


4;c2  —  83:c  + 


41' 


!081 


1073 


Extracting  the  square  root, 


2x 


whence  we  find 


41  _     /ior3  _  ViCnS 

li  -^  V  4:  ~  ~~^r~'' 

41  +  \/ior3 


X  = 


the  same  result  as  before, 

EXERCISES. 

Reduce  and  solve  the  following  equations 

X  -f  2      X  —  2 


I. 


T) 


x  —  2       x-\-2        q' 
13 


.?/  +  4      .y -^4  _  10 

^'  y  _  4  +  ^  -M  ~  y ' 


I! 


178 


q  UADRA  riG  EQ  UA  Tio:is. 


3- 

13           4 
x  —  i'^x  —  -Z~'6 

4- 

yi  _  '>ay  +  a^  —  i'  =  0. 

5- 

1                 111 

a  -\-  h  -{-  X  ~  a       b       x 

r. 

(fi       ,        h               h 

x^  —  a*      X  -\-  a      X  —  a 


-  0. 


1  + 


X  jr  a 
X  —  a 


X  —  a 


=  3. 


8. 
9- 

lO. 


1  — 

X  +  a 

__3 ?/  _ 

?/  +  a      y  —  a 


+ 


=  4. 


'2  -  y 

1 L_+_i_. 

//  —  rt       y  ■]r  a       y  —  a      y^  —  d^      y  —  a 

X  X 


a  -\-  X      a  —  X 


+  3 


0. 


PROBLEMS. 

1.  Find  two  numbers  such  that  their  difference  shall  bo 
G  and  their  product  507. 

2.  Tlie  difference  of  two  numbers  is  6,  and  the  difference 
of  their  cubes  is  93(5.     What  arc  the  numbers  ? 

3.  Divide  the  number  34  into  two  such  jiarts  that  the 
sum  of  their  squares  shall  be  double  their  product? 

4.  The  sum  of  two  numbers  is  (!0,  and  the  sum  of  their 
squares  1872.     What  are  the  numbers  ? 

5.  Find  three  numbers  such  that  the  second  shall  be  5 
jjrcater  tlian  the  lirst,  the  third  double  the  second,  and  tiie 
sum  of  their  squares  12'25. 

6.  Find  four  nundiers  such  that  each  shall  be  4  greater 
than  the  one  next  smaller,  and  the  product  of  the  two  lesser 
ones  added  to  the  product  01  the  two  greater  sludl  be  312.  \ 

7.  A  shoe  dealer  bought  a  box  of  boots  for  8210.  If  there 
had  been  5  jiair  of  l)oots  less  in  tlie  1)0X,  they  would  have  cost 
him  81  per  ])air  more,  if  he  had  still  paid  $210  for  the  whole. 
How  many  pair  of  boots  were  in  the  1)0x1:' 

Ukm.     If  wf  ciill  £  the  number  of  pairs,  the  price  paid  for  each  pair 

210 
must  have  been  - —  • 

X 


P 


4 


qUADRAriG  EQUATIONS. 


170 


> 


8.  A  linckstcr  Imufrht  a  cortuin  mimbor  of  chickens  for 
),  and  a  number  of  turkeys  for  -^  15.75.     There  were  -i  more 

chickens  than  turkeys,  l)ut  they  each  cost  him  35  cents  a  piece 

less,     llow  many  of  each  did  he  buy? 

g.  A  farmer  sold  a  certain  number  of  sheep  for  f^-lO.  If 
he  liad  sold  a  number  of  slieep  li  greater  for  tlie  same  sum,  he 
would  have  received  ^-i  a  piece  less.  How  many  sheep  did  he 
sell  ? 

10.  A  party  having  dined  together  at  a  hotel,  found  the 
bill  to  be  80.00.  Two  of  the  number  having  left  before  pay- 
ing, each  of  the  remainder  iiad  to  j)iiy  'ZA^  cents  more  to  make 
lip  the  loss.     Wiiat  was  the  number  of  the  party? 

11.  A  pcdler  bought  $10  worth  of  apples.  30  of  them 
proved  to  be  rotten,  but  he  sold  the  renuiinder  at  an  advance 
of  2  cents  each,  and  made  a  profit  of  83.20.  Ilow  many  did 
he  buy  ? 

12.  In  a  certain  number  of  hours  a  man  traveled  48  miles  ; 
if  he  had  traveled  oir'  mile  more  ])erhour,  it  would  have  taken 
him  4:  hours  less  to  ])erform  his  journey  ;  how  many  miles  did 
he  travel  per  hour  ? 

13.  The  jierimeter  of  a  rectangular  field  is  IGO  metres,  and 
its   ;irea    is 
breadth  ? 


1575   s(piare   me  ires.     What  are  its  length   and 


14.  The  length  of  a  lot  of  land  exceeds  its  breadth  by 
a  feet,  and  it  contains  m^  square  feet.  What  are  its  dimen- 
sions? 

15.  A  stage  leaves  town  A  for  town  B,  driving  8  miles  an 
hour.  Three  hours  afterward  a  stage  leaves  B  for  A  at  such  a 
sj)ced  as  to  reach  A  in  18  hours.  They  meet  when  the  second 
has  driven  as  many  hours  as  it  drives  miles  per  hour.  What 
is  the  distance  between  A  and  B  ? 

Note.  The  solution  is  very  i^^^imple  when  the  proper  quantity  is  taken 
as  unknown. 


Equations   which    may  ho   Rotlueed   to   Quad- 
ratics. 

li>(>.  Whenever  an  eqnation  contains  only  two 
powers  of  the  unknown  (jnantity,  and  the  index  of  one 
power  is  double  that  of  the  other,  the  equation  can  he 
solved  as  a  cxuadiatic. 


-  w 


j; 


1^ 


180 


QUADItATIG  EQUATIONS. 


Special  Example.    Let  us  take  the  equation 

a^  -\-  bjp  +  c  =  0. 

1 

4" 


(1) 


Transposing  c  and  adding  ^i'  to  each  side  of  the  equation, 
it  becomes 


The  first  member  of  this  equation  is  a  perfect  square, 
namely,  the  square  of  r'  +  -  ^.  Extracting  the  square  roots 
of  bo  til  members,  we  have 


a^ 


Hence, 

Extracting  the  cube  root,  ^ye  have 

X 


=  ^J-b±V{b'^-4c)]K 


General  Form.    We  now  generalize  this  solution  in  the 

following  way.     Suppose  we  can  reduce  an  equation  to  tlie 

ibrm 

ax^^  +  bx-^  +  c  =  0, 

in  which  the  exponent  n  may  be  any  quantity  whatever,  entire 
or  fractional.     By  dividing  by  a,   transposing,  and  adding 

1  Z»2 

-7  -,  to  both  sides  of  the  equation,  we  find 

4a^  ^ 


1*2       1  ^3 


a  -ia^       4fH      a 

The  first  side  of  this  equation  is  the  square  of 

lb 


x""  + 


^a 


Hence,  by  extracting  the  square  root,  and  reducing  as  in 
the  general  equation,  we  find 


x^ 


f 


■ 


t 

f » 


i 


QUADRA  TIC   KQ  IT  A  TI0N8. 


181 


(1) 


ff 


Extnacting  the  m'*  root  of  both  sides,  wc  liavo 


'. — \  I 


'{- 


If  the  exponent  ti  is  a  fraction,  tiie  sumo  course  may  bo 
followed. 

Suppose,  for  example, 

ax^  4-  bx^^  +  c  =  0. 
Dividing  by  a  and  transi)osing,  wc  have 


4 


b  2 


x«  4-     x^  = 
a 


c 

—  • 

a 


1? 
Adding   .  „  to  botn  sides, 


4        5    2    ,     Z/2  ^2        c 

The  left-hand  member  of  this  eqnation  is  the  square  of 

Extracting  the  square  root  of  both  members, 


■whence. 


M 


liaising  both  sides  of  this  equation  to  the  J  power,  wc  have 

['-h±  (/y2  -  Aw)l' 


X  = 


L 


'Za 


EXERCISES. 

1.  Find  a  number  whicb,  added  to  twice  its  square  root, 
will  make  99. 

2.  What  number  will  leave  a  remainder  cf  99  when  twice 
its  square  root  is  subtracted  from  it. 


IJ 


■ «!  -^  f  I 


a  ,1 


m 


\f 


M 


u 

)} 

'.  '.i 

i  . 

;' "i 


182 


Q  UA  DRA  TIC  EQ  U.  1 TT0N8. 


3.  Onc-nftli  of  !i  certain  nuinljcr  oxcecils  its  sqnaro  root 
l)y  ;}(>.     What,  is  tlic  niniilKT? 

4.  \\'\\\i\  number  added  to  its  square  root  nuikes  'MM\  ? 

5.  If  I'roin  3  times  a  certain  iinml)er  wo  subtract  10  times 
its  s(iuaro  root  and  00  more,  and  divide  the  remainder  by  the 
number,  tlie  (|Uotient  will  be  "Z.     What  is  tlie  number? 

Solve  the  e([uations: 


6.     3i/'-^//«  =  15. 
8.     5^-^  —  ^'i  =  13. 


7.   dif  -  7^2  =  ro. 


m 


m 


11)7.  When    the   unknown  (juantity  appears  in  the  form 

^^  ■\-    J)  the  S(piare  may  be  completed  by  sim^dy  adding  Z  to 

^  .  1     . 

this  expression,   because    a;'  +  3  H — ^    ^^   '^    perfect    sfpuire, 

1  '*' 

namely,  the  scjuare  of  x  -\-  -•    The  value  of  x  may  then  bo 

deduceil  from  it  by  solving  another  quadratic  equation. 


EXAMI'LE. 


3a;2  +  --,  =  22. 
x^ 


We  first  divide  by  3  and  add  2  to  cacli  side  of  the  equation, 
obtaining  1         03  ^8 

^^  +  ^  +  :^^  =  ir  +  ^  =  y 

Extracting  the  square  root  of  both  sides, 

1        2V7       2\/21       3    .,, 

By  multiplying  by  x,  this  equation  becomes  a  quadratic, 
and  can  be  solved  in  the  usual  way. 

Let  us  now  take  this  equation  in  the  more  general  form. 


1 

X  -{-  -  =  r, 

X 


{«) 


2 


which  reduces  to  the  foregoing  by  putting  c  =     \/21.    Clear- 
ing of  fractions  and  transposing, 

x^  —  c.r  4-  1  =  0  ; 


) 


ri 


M 


I 


I 


i 


Vi 


i 


QUADRA  rrc   KQVATIOXS. 

which  being  solved  in  thc^  usuiii  way,  gives 

__  c  ±  \/{c^  —  4) 


183 


X 


The  two  roots  arc  therefore 

^1  -  jj  , 

—     , ^   —  ~    "       — • 

If*  in  the  first  of  these  e(inalions  we  riitioiializc  the  niimor- 
iitor  by  multiplying  it  by  e  —  V{e^  —  4)  (§  185),  we  ahull  lintl 


.r,  = 


it  to  red  nee  to 


e- V(t'=^-4) 


that  is,  to      -    There  fore, 


X 


a:^  =     -  iilentiralhj. 


Vice  verm,  x^  is  identically  the  samo  as  --• 
This  must  be  the  case  whenever  we  solve  an  equation  of  the 
form  (a),  that  is,  one  in  w^hieh   the  value  of  x  +      is  given. 


50 


Let  us  suppose  first  that  e  =  '  -,  so  that  the  equatiun  is 

1        50 

X  +   -  =  -  . 
X        7 

It  is  evident  that  x  =  7  is  a  root  of  this  equation,  because 
Avhen  Ave  put  7  for  .r,  the  left-hand  member  becomes  7  +  -, 

50  1 

wliic'h  is  equal  to  y    If  we  put  ^  for  x,  the  left-hand  mem- 
ber will  become 

7  ^    1    ~  7  ^  ^* 

Hence  x  and      exchange  values  by  i)utting  ^  instead  of  7, 

X  / 

so  that  their  sum  x  +      remahis  unaltered  by  the  change. 


I 


J,. 


1^  I 


184  QUADIiATIC    EQUATIONS. 

The  gcncrul  result  imiy  bo  expressed  thus: 

Ueettuso  tlio  vfiluo  of  the  exiiressioii  x -V-      rormiins  un- 

1  ^" 

altered  when  we  chaii^^e  ./;  into     ,  therefore  the  reciproeul  of 

uiiy  root  of  the  equatiou 

1 

X  +       =  (5 
X 

is  also  u  root  of  the  same  equation. 


,1 


IN  i'ji  1 


w  li 


EXERCISES. 

Find  all  the  roots  of  the  followin*;  equations  witliout  elear- 
in<,^  the  given  oijuations  from  den(>niinal(»rs: 

1    _  17 


I.  :i-3  +  :.  = 


2.     a'x"  +  -:,  7.  =  in^  —  JJ. 


a'x^ 


3.     in/-f]^o  =  28. 


.  ^''       .        2 

4-    -:f  +  f 


2^2. 


5.  Show,  without  solving;,   tliat  if  r  he  any  root  of  the 


equation 


■2-t.  _   — 


x^  + 


x^ 


a. 


then  —  r,     ,  and  —      will  also  be  roots. 
r  r 


Factoring  a  Quadratic  Equation. 
198.    1.  Special  Case.     Let  us  consider  the  equation 


x" 


32:  —  15  =  0, 


or 
or 


a;2  _  2.C  +  1  —  10  =  0, 

{x-iy-4:^  =  0. 

Faetoriiig,  it  Ijccomes  (§  90), 

{x  -1  +  4)  {x  _  1  _  4)  r=  0, 
or  {x  +  3)  {x  —  5)  =  0. 

Therefore  the  original  equation  can  be  transformed  into 
{x  +  3)  {x  -  5)  ^  0, 

a  result  which  can  be  proved  by  simply  performing  the  multi- 
plications. 


g  UA  Dli.  1  no   h'Q  l/A  TIONS. 


185 


i 


Thi!-".  last,  equation  may  bo  HutisfU'tl  by  puiliiig  citluT  of  iU 
factors  c(iual  to  zoro  ;  that  is,  by  !^iii)i)()sin<^ 

m:  -f-  3  =  0,     whi'iico    X  =  —'.)  ', 
or  a;  —  5  =  0,     whuiico    ^  =  +  5. 

These  are  the  same  roots  which  we  should  ol>taiii  by  solving 
tlie  original  e^iuation. 

2.  Far/oritifj  the  Grnrrnl  Qnddm/ic  /Jqitaliun.  Let  us  con- 
sider the  general  (iuadratic  eciuation, 

a;2  ^  py  ^  q  =  0.  (tl) 

Now,  instead  of  thinking  of  x  as  a  root  of  this  C((uation, 
let  us  su})pose  x  to  luive  any  value  whatever,  and  let  us  con- 
sider the  expref<sioii 

xi  +  px  +  q,  (1) 

wluch  for  shortness  wo  shall  rail  T.     Let  us  also  inf|uiro  how 
it  can  be  transformed  without  changing  its  value. 

First  wo  add  and  subtract  .p%  so  as  to  make  part  of  it  u 
perfect  s(iuarc.     It  thus  becomes, 

X  =  x^  +  px  +  ^1)^  —  ^p"^  +  q  \ 
or,  which  is  the  same  thing, 

Factoring  this  expression  as  in  §  188,  it  becomes 
X=\x-\.\p  +  Q/>^  -  (/)-]  [x  +  \p-  (^/  -  (if\. 

Tlio  stiulpnt  sliould  now  prove  tliat  this  expression  is  really  equal  to 
a''^  +  px  +  q,  by  perforniing  the  multiplication. 

Let  us  next  put,  for  brevity, 


"  =  -.^1' 


« 


?  =  -  )it>  +  {if  - 'i)  ■ 


(2) 


.1       . 


;  1 


1 


>l     > 


18G 


Q  U.  I  Z>i?  J  TIC   EQ  UA  TI0N8. 


The  preceding  value  of  X  will  then  become, 

X  =  {x  -  ic)  {X  -  (3),  .  (3) 

an  exprepsion  identically  equal  to  (1),  when  we  put  for  a  and 
(3  their  values  in  (2). 

Let  us  return  to  the  suppositien  that  this  expression  is  to 
he  equal  to  zero,  and  that  ..■;  is  a  root  of  the  equation. 

The  equation  {a)  will  tlien  be 

{x  -  a)  {X  -  ,3)  -^  0.  (4) 

But  no  product  can  be  equal  to  zero  unless  one  of  the  fac- 
tors is  zero.     Hence  we  must  have  either 

X  —  «  =r  0,     whence     x  =  a  ; 

or  X  —  (3  =  0,     whence    x  =  l^. 

ITenco,  a  and  ^i  ai-e  the  two  roots  of  the  equation  {/(). 
Th(^   above   is    another    wav  of    solvinir    the    (Muuh'atio 
equation. 

To  com})are  the  expressions  (I)  and  (C),  let  us  perform  the 
multiplication  in  the  latter.     It  will  become, 

X  =  .i-'^  -  («  +  [3)  X  +  cf3. 

Since  this  expression  is  identically  the  same  as  x^-^px-^-q, 

the  coeliicients   of  the  like  j)owers  of  x  must  be  the  same. 

That  is, 

a -{-[3=  -p,) 

which  can  be  readily  proved  by  adding  and  multiplying  the 
equations  {2). 

This  result  may  be  expressed  as  follows  : 

]       Theorem.    When  a  quadratic  equation  is  reduced 
'  to  the  general  form 

the  coefficient  of.?.'  will  be  eciual  to  the  sum  of  the  roots 
with  the  sio-n  chaii,2:('d. 

The  term  independent  of  x  will  be  equal  to  the 
product  of  the  roots. 

The  student  may  ask  wliy  can  we  nf»t  determine  the  roots  of  the 
quadratic  ecjiiatiou  from  eijuationa  (,5),  regarding  a  uud  \3  us  the  uuknuvvn 
quantities  V 


(P 


/ 


%) 


iff  V- 


"fl 


q  UABIi.  1  TIG   EQ  UA  TIONS. 


187 


is  to 


(5) 


need 


Wo  can  do  sn,  but  let  us  see  what  tlio  result  will  be.     Wc  eliminate 
cither  «  or  /j  by  substitution  or  by  comi)ariBon. 

From  the  second  equation  (5)  we  have, 

(C 


Substituting  this  in  the  first  equation,  we  have 
Clearing  of  fractions  and  transposing. 


«  +  ?  = 


«'  +  ]h:  +  q 


0. 


Wo  have  now  the  same  equation  with  which  we  started,  only  stakes 
the  place  of  .?'.  If  we  had  elinnnated  <i,  we  should  have  had  the  sumo 
equation  in  /3,  namely, 

/33  +  pH  +  (/  =  0. 

So  the  equations  (5),  when  we  try  to  solve  them,  only  lead  us  to  the 
origiiud  equation. 

<4^  199.   To  form  a  Quadratic  Equation  wtieii  the  Roots  arc 

given.  The  foregoing  principles  will  enable  ns  to  ibrni  a  quad- 
ratic equation  which  shall  have  any  given  roots.  We-  have 
only  to  substitute  the  roots  for  ce  and  [3  in  equation  (4),  and 
perform  the  multiplications. 

EXERCISES. 

Form  equations  of  wliich  the  roots  shall  be : 
I.      -t-  1  and  —  1.  2.     3  and  2. 

4.     3  +  3^10  and  3— 2a/10. 


3 

5 
7 
9 

II 
13 

17 


;f  1  and  +  2. 
—  1  and  —  2. 

2  -j-  Vii  and  2  —  Vo. 


—  3  and  —  2. 
7  +  3a/3  and  7-21/3.  6. 

—  1  and  +  2.  8. 
+  1  and  —  2.  lo. 

3  4 
'-    and  -•  12. 

4  .) 

2+^/2  and  2  — \/2.  14.     9  +  2a/2  and  9  —  2^2. 

5-1-71/5  and  5  —  71/5.  16.     a  -^  b  and  a  —  b. 
a  -f-  Vcc^  —  b^  and  a  —  y/cfi  —  b\ 


7       ,  9 
2  ""'^  2' 


l\ 


h 


188 


IMAGINARY  ROOTS. 


\\-  V 


(I   ' 


Equations  liaviiig  Iniajyinary  Roots. 

200.  When  Ave  complete  tlio  sciuare  in  order  to  solve  a 
quadratic  c([uation,  the  quantity  on  the  riglit-lumd  side  of  tlie 
cfjuation  to  wliich  that  square  is  equal  must  be  positive,  else 
there  can  he  no  real  root.  For  if  we  square  either  a  positive 
or  negative  quantity,  the  result  will  he  positive.  Hence,  if 
the  square  of  the  first  member  comes  out  equal  to  a  negative 
quantity,  there  is  no  answer,  either  positive  or  negative,  Avhich 
\i\\\  fulfil  the  conditions.  Such  a  result  shows  that  impossible 
conditions  have  been  introduced  into  the  i)roblcm. 

EXAMPLES. 

1.  To  divide  the  number  10  into  two  such  parts  that  their 
product  shall  be  34. 

If  we  proceed  witli  this  equation  in  the  usual  way,  we  shall 
have,  on  completing  the  S(iuare, 

xi  _  lOa;  +  25  =  —  9, 
or  {x  —  5)2  =  —  9. 

The  square  being  negative,  there  is  no  answer.  On  con- 
sidering the  question,  we  shall  see  that  the  greatest  possible 
product  which  the  two  i:)arts  of  10  can  have  is  when  they  are 
each  5.  It  is  therefore  impossible  to  divide  the  number  10 
into  two  parts  of  which  the  product  shall  be  more  than  25  ;  and 
because  the  question  supposes  the  product  to  be  34,  it  is  im- 
possible in  ordinary  numbers. 

2.  Suppose  a  person  to  travel  on  the  surface  of  the  earth  to 
any  distance ;  how  far  must  he  go  in  order  that  the  straight 
line  through  the  round  earth  from  the  point  whence  he  started 
to  the  point  at  which  he  arrives  shall  be  8000  miles? 

It  is  evident  that  the  greatest  possible  length  of  this  line  is 
a  diameter  of  the  earth,  namely,  7,912  miles.  Hence  he  can 
never  get  8,000  miles  away,  and  the  answer  is  imjwssible. 

In  such  eases  the  s(|uare  root  of  the  negative  quantity  is 
considered  to  be  part  of  a  root  of  the  equation,  and  because  it 
is  not  equal  to  any  positive  or  negative  algebraic  quantity,  it 
is  called  an  iinaginary  root.  The  theory  of  such  roots  will  be 
explained  in  a  subsequent  book. 


11 


w 


a] 


it 

q 

C( 

t( 

0] 

S 
ii 


I. 

•  solve  a 
le  of  the 
Live,  else 
positive 
lence,  if 
negative 
c,  Avhicli 
1  possible 


lat  their 
we  shall 


On  con- 
possible 
they  are 
mbcr  10 


Jo  ;  and 

t  is  im- 


earth  to 
straight 
s  started 

s  line  is 
he  can 
)le. 

mtity  is 

cause  it 

iitity,  it 

will  be 


IRRATIONAL   EQUATIONS.  189 


CHAPTER    111. 

REDUCTION    OF    IRRATIONAL    EQUATIONS    TO    THE 

NORMAL    FORM. 

201.  An  Irrational  Equation  is  one  in  which  the 
unknown  quantity  appears  under  the  radical  sign. 

An  irrational  equation  may  bo  cleared  of  fractions 
in  tlie  same  way  as  if  it  were  rational. 

Example.    Clear  from  fractions  the  equation 
^/x  -\-  a  -\-  "s/x  —  a  2a 


Va-'  -\-  a  —  V- 


X 


a 


^0? 


cv" 


Multiplying  both  members  by  V^'^  —  <^^  =  'Vx-\-a  \/x—a^ 
we  have 


{x  4-  ii)  'Vx  —  rt  -f  {x  —  a)  ^/x  +  a 


=:  2«. 


a. 


^/x  -\-  a  —  ^/x  —  a 
Next,  multiplying  by  Vx  +  «  —  V^  —  (t,  we  have 

{x-\-a)  '\/x  —  a  -\-  [x  —  a)  V'J^  +  «  =  2rt\/^i;4-rt  —  2aVx 
Transposing  and  reducing,  we  have 

(x  +  3a)  Vx  —  «  +  {x  —  da)  \/x  +  «  =  0, 
and  the  equation  is  cleared  of  denominators. 

Clearing  of  Surds. 

-  *203.  In  order  that  an  irrational  equation  may  be  solved, 
it  must  also  bo  cleared  of  surds  which  contain  the  unknown 
quantity.  In  showing  how  this  is  done,  we  shall  su})pose  the 
equation  to  be  cleared  of  denominators,  and  to  be  composed  of 
terms  some  or  all  of  which  are  multiplied  by  the  square  roots 
of  given  functions  of  x. 

Let  us  take,  as  a  first  example,  the  equation  just  f  on  nil. 
Since  a  surd  may  be  either  positive  or  negative,  the  equation 
in  question  may  mean  any  one  of  the  following  four : 


t 


190 


IRRATIONAL    EQUATIONS. 


M  i! 


i( 


w 


{x  +  3r/)  ^/x~^i  +  {x  —  "da)  ^x  +  a  =  0,  (1 ) 

(./:  +  ^(()  a/.^-^  —  {x  —  oa)  ^/x  +  a  =  0,  (:i) 

—  (.r  +  ^(f)  Vx  —  a  +  {x  —  3r0  V^+~^  ==  0,  (:]) 

—  {x  +  da)  Vx~—  a  —  {x  —  :}(()  ^/.^•  +  a  —  0.  (I) 

But  the  third  eqiiution  is  merely  tlie  negutive  of  the  second, 
and  the  fonrth  the  negative  of  the  lirst,  so  that  only  two  have 
dill'crent  roots.     Let  us  jjut,  for  brevity, 


P  =  {x  +  '6a)  Vx  —  a-{-  {x  —  3a)  Vx  + 


7,| 

a,  ) 


(5) 


(^') 


Q  =  (x  -i-  da)  Vx  —  a  —  {x  —  3a)  Vx  + 
and  let  us  consider  the  equation, 

PQ  =  0, 

Since  this  equation  is  satisfied  when,  and  only  wlien,  we 
have  either  P  =iO  or  Q  =  0,  it  follows  that  every  value  of  x 
which  satislies  either  of  the  equations  (1)  or  (2)  will  satisfy  {(>). 
Also,  every  root  of  (G)  must  be  a  root    ither  of  (1)  or  {%). 

If  we  substitute  in  (G)  the  values  of  P  and  Q  in  (5),  we 
shall  then  have 

(x  +  3^0^  {x  —  a)  —  {x  —  3a)^  {x  +  a)  =  0, 

■which  reduces  to  5x^  —  Oa^  =  0, 


and  gives 


,    3a 

X  =  ±  ----- 

V5 


It  will  be  remarked  that  the  process  by  which  we  free  the 
equation  from  surds  is  similar  to  that  for  rationalizing  the 
terms  of  a  fraction  emj)loyed  in  §  185. 

As  a  second  example,  let  us  take  the  equation, 

Vx  +  11  +  Vx'^^i  —  5  =  0.  (a) 

We  write  the  three  additional  equations  formed  by  combin- 
ing the  positive  and  negative  values  of  the  surds  in  every  way : 

—  Vx  +  11  +  Vx  —  4:  —  5  =  0, 
Vx^ii  —  Vx  —  4  —  5  =  0, 

—  Vx  +  11  —  Vx  —  4  —  5  =  0. 
The  product  of  the  hrst  two  equations  is 


0 


in  UA  TIONA  L    EQ  UA  TIOXS. 


191 


0) 
('0 
{^) 

(0 

second, 
vo  have 


(5) 


'hen,  we 
luc  of  X 
tisfv  (0). 

(2): 

L  (5),  we 


free  the 
m\g  the 


(^0 

eom])in- 
jry  way : 


i 


i 


(V.i--4-5)2-(.r+ll)  =  0, 

or  10  —  lOV.r^l  =  0.  (1) 

The  product  of  tlic  lust  two  is 

10  +  10a/.a:-1  =  0.  (3) 

The  product  of  these  two  products  is 

100  -  100  {x  —  4)  ==  0, 
Avhich  gives  x  =  5. 

It  will  be  remarked  that  (:2)  differs  from  (1)  only  in  having 
the  sign  of  the  surd  different.  This  must  be  the  case,  because 
the  second  pair  of  ecjuations  formed  from  {(()  differ  from  the 
first  pair  only  in  having  the  sign  of  the  surd  \/x  —  4  different. 
Hence  it  is  not  necessary  to  write  more  than  one  pair  of  the 
equations  at  each  step.     The  general  process  is  as  follows : 

I.  Change  the  fi7<^ii  of  one  of  the  surds  in  the  giveiv 
cqitatioii,  and  multiply  tlie  equation  thus  fonned  hy  the 
ori^'inal  equa tion. 

II.  lleduce  this  ])roduet,  in  it  change  the  si^n  of  an- 
other of  the  surds,  and,  form  a  new  product  of  the  two 
equations  thus  formed. 

III.  Continue  the  process  until  an  equation  without 
surds  is  reached. 

Example.     Solve 

\^^x  +  9  +  \/-lx  +  G  +  V-c  +  1  =  0. 
Changing  the  sign  of  ^/x  -\-  4, 


VS^  +  *)  +  ^'Zx  +  G  —  \^x  4-4  =  0. 
The  product  is 

{\/^x  +  y  +  \/-Zx  +  (jf  —  (.-6-  +  4)  =  0, 
or,  after  reduction, 


9.1;  +  11  +  2  a/8.1-  +  9  V'lx  +  G  =r  0. 
Changing  the  sign  of  's/'ix  -f-  G,  we  have 

9.1-  +  11  —  2\/8T+l)  V'^x  +  C  =  0. 


I 


192 


IliliA  TIONA  L   EQ  UA  TIONS. 


il    >i 


iJ    ^ 


Tlic  product  of  the  last  two  equations  reduces  to 
17u;2  —  (j(jx  —  95  =  0, 

33  ±52 


■\vliicli  belli <?  solved  ffives 


X  = 


'b  ""• '  ^^^  C3 '  ■  ^"  ~  —         -|  «/ 


Rkmark.  Equations  containing  surds  may  often  roduce  to  the  form 
treated  in  §  106.  lu  this  case,  the  methods  of  that  section  may  be  fol- 
lowed. 

EXERCISES. 

Solve  the  equations : 


I. 


2. 


'        +        ' 


2Va  —  2V^ 


V^  +  Va       Vx  —  Va 

Vx^  +(1  X 


X  —  a 
3.     y/x  ^'6  —  ViC  — 4  =  1. 


4.  v^^  n  +  Vx- 14  =  14. 

5.       (3_2;)l_(3+2;2)]   _  Q. 

7    __L_    .  jv/^ L__ 

'■  \/5.r  +  3  2 


=  0. 


).    V^i^  —  2-i'*  -i 


a; 


10. 


II. 


X  -\-  Vx  _  .T(a:  —  1) 
'x^^x  ~         4 


-^  =  b. 


Vl-^a 


'\/x  ~  a  4-  Vitx  —  1       V^  —  1 


H  I 


SIMVLTANEO US   Q UADRA TKJ  EQ UA TIOXS.        VX] 


le  form 
be  fol- 


=  1. 


I 


CHAPTER    IV. 

SIMULTANEOUS    QUADRATIC    EQUATIONS. 

Between  a  pair  of  sinuilhmoous  general  qiiatlrulie  erinatious 
one  of  the  unknown  quantities  can  always  be  eliminated.  The 
resultiiif,'  equation,  when  redueed,  will  l)e  of  the  fourth  degree 
uith  respect  to  the  other  unknown  quantity,  and  cannot  bo 
solved  like  a  (juadratic  etpiation. 

But  there  are  several  cases  in  which  a  solution  of  two  equa- 
tions, one  of  which  is  of  the  second  or  some  higher  degree, 
may  be  effected,  owing  to  some  of  the  terms  being  wanting  in 
one  or  both  equations. 

20.3.  Case  I.  IV/ten  one  of  the  equations  is  of 
the  Jii'st  def/ree  onlt/. 

This  case  may  be  solved  thus  : 

IiULE.  Find  the  '.uluo  of  aim  of  the  unlcnmiii  quan- 
tities ill  terms  of  the  other  from  the  eqiintioii  of  the  frst 
decree,  lliis  value  heinij  siihstituted  in  the  otlter  eqaa- 
tion,  ice  shall  have  a  qa a dr(( tic  equation  from  wldch  the 
other  unknown  quantity  niaij  he  found. 

Exa:mple.     Solve 

22-2  +  3.i-^  —  5?/2  —  X  —  5?/  =  2G,  I 

22'  —  3^  =    5.  I 
From  the  second  equation  we  find 

3?/  +  5 


(«) 


X  = 


2 


(^>) 


Whence, 


.2  _  0//^  +  30//  +  25 


x^  = 


Substituting  this  value  in  the  first  equation  and  reducing, 
we  find 

4//2+ lOy +  10  -  20. 

Solving  this  quadratic  equation, 
13 


194        SIMULTANEOUS   'iUADRATIG  E'^UAriOSS. 

,j  =  —2  ±V^  =  —2±  2V2. 

This  value  of  ?/  being  substituted  in  the  c(iuiiiion  (/y)  gives, 

—  1  ±  3V8  _  —  1  ±  (\V2 
2     ~  "  'Z 


X  •=. 


The  eaiTi(!  problem  may  be  solvi'd  in  the  reverse  order  by  eliminating 
y  iustead  of  x.    The  second  (Miuation  (11)  gives 


y 


2x  —  5 


If  we  substitute  this  value  of?/  in  the  first  equation,  we  shall  have  a 
quadratic  equation  in  x,  from  which  the  value  of  the  latter  (juautity  can 
be  found. 


Solve 

I. 

2. 


EXERCISES, 


X' 


t;2  _  2X1/  +  4?/2  =   21. 

2x  +  y  =  13. 

3^3  _  0^2  _^  5.^,  _  2,/  =  28. 
a;  +  ?/  +  -i  =  0. 

5.r^  +  ;i/2  -  X  -  ij  =  72, 
X  +  2ij  =  0. 

3:c2  +  2jf  =  813, 
7a;  -  4^  =    17. 


X 

+  y  = 

I 

X 

y 

7 

y 

X 

12 

304.  Case  II.  When  eacJi  equation  contains 
only  one  term  of  the  second  dcfjt'eef  and  that  tenn 
has  the  same 2>t*od act  or  square  of  the  unknoivn 
quantities  in  the  two  equations. 

Such  equations  are 

ax'^  +  (Ix  +  ry  +  /  =  0,  ] 
a'x^-\-d'x  +  e'y  -f/  =  0,  f 

where  the  only  term  of  the  second  degree  is  that  in  x^. 

If  we  eliminate  x^  from  these  equations  by  multiplying  the 
first  by  a'  and  the  second  by  a,  and  subtracting,  we  have 


(^0 


.1 


4 


SIMULTANEOUS   QUADRATIC   EQUATIONS.         195 


J))  gives, 


liminatiug 


lall  liave  a 
[uuutity  can 


that  term 
unhnoivn 


(^0 


m  0"= 


\v 


ltq)lyiiig  the 
,e  have 


i 


{(I'd  —  wl')  X  +  {a'c  —  ac')  y  +  a'f  —  nf  —  0. 

Solving  this  i'(|iuitioii  with  ivspcct  Lo  ./•,  we  liiul 

{ac  —  a'c)  11  +  (if  —  a'f 
a'd  —  ad' 


X 


(») 


(") 


By  siihstitutlnpj  this  vahiu  of  x  in  oilliorof  tlic  equations 
{((),  wu  !?h;ill  iiave  a  (lUiidratiu  ui|Uation  in  //.  Solvin;::  tiic 
latter,  we  shall  obtain  two  values  of  y.  .Substituting  these  in 
(/>),  we  sliall  have  tlie  two  corres})onding  values  of  .r,  and  the 
sohuion  will  be  com})lete.     Hence  the  rule, 

Kruninate  the  term  of  thr  second  dc<Jvee  hi/  addttioii 
or  siihtracHon,mLd  use  the  resiilthi£  e(/u(itioii  of  the  Jlrsb 
degree  with  either  of  the  original  equations,  as  in  Case  I. 

Example.     Solve 

Zcy  —  4:X  +  5//  —  23, 
3x?j  +  T.f  +    !/  =  -i^L 

Multiplying  the  first  equation  Ijy  o  and  the  second  by  2, 
and  .nibtraeting,  we  have 

—  2Gx  +  13//  =  —13;  (L) 

1  1 

whence,  x  =  ,^i/  -\-  ^^-  (r) 

Substituting  this  value  in  the  first  equation,  we  find  a 
quadratic  equation,  which,  being  solved,  gives 

y  =  —2  ±  V  29. 

Substituting  these  values  in  (c),  the  result  is 

x=-l±lV-Z9. 

The  two  sets  of  values  of  the  unknown  quantities  are 
therefore 


^1  =  -^+2^-^' 

7/i  =  -  2  +  V2d, 


'<-2    —    —  .)   —  9  Y'vJ. 


1/2 


/v   V    -vt/. 


We  might  have  obtainod  tho  same  result  hy  solving  the  ofjuation  (e) 
with  respoct  to  ?/,  and  substituting  n\  {a).  The  student  should  practice 
both  methods. 


;■■  v: 


mm 


m 


wi^* 


lU 0      siMUL tam:o us  q ijadka  tic  i:q  ua  noiXti. 


w 


I. 


2. 


EXERCISES. 

C/i  _  'Ar  _  .\,f  —  ^)5, 
a'^  4-  :i.r  —  :\//  =  is. 

2//-'  -f  //  =  '2S. 
y'i  +  :\c  —  4//  =  11 

3.  .ry  4-  f;.f  -f  7//  =  (ill, 

905.  Cask  III.  W/icn  iteifJiPV  rquaffon  coii^ 
tdliiH  a  term  of  t/ie  first  dcf/rec  hi  x  or  //. 

Hulk.  Fliminate  the  constant,  terms  hij  vinltijihjin'J 
Cdcli  ('(lUdtioii  hi/  t/ie  (U)nst((iit  term,  0/  (iir  oilier,  iind 
(nl(lin<J  or  srthtraetino'  flie  firo  products.  Tlie  result  will, 
tjo  a  (juiulrat'ie  efjUdtlon,  J'roDi  which,  ci titer  itnknon.'ih 
qnnniitii  cirn.  1)e  deterniincd  in  terms  of  the  other.  Then 
suhstitiite  as  in  Case  I. 


ExAMi'LE.     Solve 

14  X  1st  C(i., 

5  x2d  cq., 

SuLtnicLiMg-, 


x'+    .ry/-    7/2 


5,1 


0! 


(1) 


4.i:2  +  ;20.r//  —  2\u'^  = 

This  is  a  qiiadnitic  equation,  1)y  which  one  iniknowii  (|iiiiii- 
tity  can  he  expressed  in  terms  of  the  other  without  the  latter 
heing  under  the  radical  sign. 

Transposing,  4.v^  +  29.r?/  =  24//-. 


(2) 


Comideting  square,    ix^  -f  20.zij  +  -.tt^/^  =  'v-'^^' 


Extracting  root, 


Whence, 


29  ,   35 

2x-i--j7/  =  ±j-!/' 

—  20  ±  35  3 


Suhstituting  the  fir.^t  of  these  values  of  x  in  either  of  the 
original  equations,  we  shall  hayo 


y' 


IGj 


l\ 


)tl   VOU" 

• 

Uiplijiii'J 
her,  (did 
•HitU  trill 
uiiknnirih 
I'V.     Then 

\       (0 


own  (|Uim- 
tho  lalter 


i^ 


,.-r 


oa 


or  —  8//. 
.tlicr  of  the 


I 


■^ 


SIMULTANEOUS   QUADliATIC   KqCMlONS,         107 

whence,  y  =  ±  -^ ;        x  =  ±,X 

JSiibstiLiitiug  Ihe  second  vahic  of  x,  wo  have 

,        1 


whence, 


.1  -T-     8 


'riicrcforc  thi'  four  p().ssil)lo  vtihics  of  the  unknown  qnanti- 
ties  urc, 

Vu         Vu 
1  1 

?/=+!,     —4,     —  - --,      +     7— • 

E:ich  of  (liL'so  four  [)airs  of  vahics  saiislk'S  the  original 
Cfjuadoii. 

A  slight  change  in  tlio  mode  of  proceeding  is  to  (livi(U'  llio 
equation  {'I)  I)y  eiflier  /'  or  //■,  and  to  Ihid  tlic  vaUic  of  the 
quotient.     Dividing  by  ij'^  and  putting 

X 

y 

tlio  equation  will  hecomo 

42^2  4-  20?^  —  U  =  0. 
This  ([uadratic  equation,  being  solved,  gives 
—  20  ±  -i'")        3 


11 


=  -^or  -8, 


X 


Putting  *    for  n,  and  multij)lying  by  y, 
If 

3 


Solve 
I. 

2. 


X  =   .  >j  or  —  8?/,  as  before. 

■i' 


EXERCISES, 


.tS 

— 

,T_y  -1- 

f 

— 

3 

— 

0, 

a;2 

— 

2.///  + 

•if 

— 

4 

— 

0. 

2x^ 

+ 

:!.r//  - 

f 

— 

z= 

0, 

.r2 

+ 

■If 

+ 

1 

= 

0. 

I'  IJ 


!"  ' 


r, 


') '  il 


198        SnnrLTANKOUS  qUADHATIG   KQUATIONS, 


Pi 


*^(>(>.  Casi:  IV.  WIhii  tlie  vj'in'vssions  contain-' 
lnf/  t/ir  itn/»'noirn  <jiiantiticH  in  the  two  vquationM 
/tdcc  coimnoit  J'ac/ors, 

JiiM:.  Dii'idr  one  of  the  r(/nft/ions  irhich  Cf(/h  he  Jar- 
tovcd  Ifij  the  otlicr,  tnul  canrrl  the  cnmmoii  J'(t('ttirs. 
Then  clear  of  frnct'wns,  if  neeessavij,  and  we  shall  have 
an  equation  of  a  lower  decree, 

EXAMPLES. 

1.  .r3  +lf  =  01,      X  -\-  1J  z=  7. 

Wc  hiivc  seen  (5^  04,  Th.  1)  tlitit  x^-\-if  is  aivisil)lo  by  z  +  t/. 
So  dividing  tlio  first  tHiiuitioii  by  tlio  second,  we  hiivo 

a;3  _  ry  +  y'  =  13. 
This  is  nn  equation  of  the  second  degree  only,  and  wlien 
combined  witii  tiie  second  of  the  original  e(|Uiitions,  the  solu- 
tion may  be  ell'ccted  by  Case  I.     The  result  is, 
a*  =  3  or  4,        i/  =  4  or  3. 

2.  xy  1-  ?/-  =  133,    r?;2  —  if  =  05. 

Factoring  the  first  member  of  each  equation,  the  equations 

Ijccomc 

1/  (x  +  1/)  =  133,        (X  +  y)  {x  -y)  =  05. 

Dividing  one  equal  ion  by  the  other,  and  clearing  of  fractions, 

7 
V2y  =  7x,    or    y  =  ~x. 

The  problem  is  now  reduced  to  Case  I,  this  value  of  y 
being  combined  with  either  of  the  original  equations. 

207.  There  are  many  other  devices  by  Avhich  simultaneous 
equations  may  be  solved  or  brought  under  one  of  tiie  above 
cases,  for  which  no  general  rule  can  be  given,  and  in  wliicli 
the  solution  must  be  left  to  the  ingenuity  of  the  student. 
Sometimes,  also,  an  equation  which  comes  under  one  of  the 
cases  can  be  solved  much  more  expeditiously  than  by  the  rule. 

Let  us  take,  for  instance,  the  e(piations, 

.7;2  +  y^  =  G5,        xy  =  28. 

These  ec|uations  can  be  solved  by  Case  III,  but  the  work 
would  be  long  and  cumbrous.     We  see  that  by  adding  and 


\ 


1 


SJML'LTAyiSOUS   QUAIJILITIC   hQirATIONS. 


100 


nfd  fli- 
nt ions 

\  he  f(ic- 
ill  hare 


h^x+ij. 


lul  vlicn 
the  sulii- 


'qiuiUou3 


fractions, 


luc  of  y 

lUancous 
10  iil)ove 
in  wliich 
-tiuloni. 
10  (>r  the 
the  rule. 


tlic  work 
ling  und 


! 


\ 


8ul)traf'tiiif]r  twice  tlio  Rccond  ciinntion  to  uiul  from  tlio  first, 
wt"  Clin  form  two  itcrlect  ^(inarcs.  Kxtructing  the  ntots  ul 
tlicsi'  s(iuarcs,  wo  slnill  have  two  simple  conations,  whicli  shall 
^ivc  the  solution  at  once.  Kadi  unknown  (|uantil y  will  havo 
four  values,  namely,   ±  7  ±  4. 

FHODLEMS    AND     EXEHCISES. 

Tli(>  following?  t'(]uutionH  can  all  bo  solved  by  boiuc  Hlu)rf,  and  cxpo- 
ditious  ('(iinbiiiatioM  of  tin-  ('(nr.itioiis,  or  by  factoiiiij;,  witlioiit  ^'oin^f 
fliruii^li  tlif  complex  i>|i>c('SH  of  ("awe  III.  'I'lie  .sfiKJeiil  is  n-coimiieiidcd 
not  to  work  upon  the  e(|iiatioii.s  at  random,  but  to  study  eaeb  pair  until 
he  sees  bow  it  can  bo  reduced  to  n  simpler  e(]untlon  l)y  addition,  miilii- 
l)iication,  or  factoring,  and  then  to  go  tlirougli  thu  operations  thus  sng- 
gesteil. 

1.  if  -\-  ry  =  14,     tr'  +  r 7/  =  ,'}5. 

2.  4a;3  —  ^Lry  =  208,    2xi/  —  f  =  39. 

3.  x^  +  ij  =  4x,    f  +  a;  =  4^. 

If  wo  subtract  ono  of  theso  (Hiuatious  from  the  other,  the  dilTennico 
will  be  divisible  hy  x  —  1/. 

4.  .r3  +  2/3  +  3.r  4- 3// =  378,    x^  +  f —  3x  —  liy  =  ,}U. 


5- 
6. 


8. 

9- 
10. 

II. 


xi  +  if  =  ^4,    x  +  y  =  VZ. 

x^  -\-  xy  =  03,    a;3  —  ?/3  =  77. 

^/x  +  Vy  _ 

y/x  —  Vy 

x^  +  xy  =  a,    y"^  -f  xy  =  b. 

x^  -\-  xxf  =  10,    y^  ^  x?y  =  5. 


4,     a-2  —  7/2  —  544^ 


x  —  aVx  -\-  y,    y  ■=■  h\/x  -\-  y. 

x\/x  -{-  y  =  12.     yVx  +  ?/  =  15. 

12.  2x'  -f  2y'^  =  X  ■\-  y,     x^  ■\-  y^  =  x  —  y. 

13.  5.r''  —  5_?/2  =z  x  -{■  y,     ox^  —  oy"^  =z  x  —  y. 

14.  .r2  +  ?/2-f-.^2  _  30,     xy-tyz+zx  =  17,     x  —  y  —  z  =  2. 

15. 


•'  \  x  —  y       x  —  y 


200 


1    lODLEMS. 


ji 


1 6.  A  principal  of  -^5000  amounts,  'wifli  simple  interest,  <o 
$7100  after  ii  certain  nun/ocr  of  years.  Had  tliu  rate  of  inter- 
est been  1  percent,  liiglierand  the  time  1  year  longer,  it  would 
have  amounted  to  -^7800.     What  was  the  time  and  rate? 

17.  A  courier  left  a  station  ridinpr  at  a  uniform  rate.  Five 
hours  af'terwai'd,  a  second  I'ollowL'd  him,  riding  3  miles  an 
hour  faster.  Two  hours  after  tlie  secoiul,  a  third  started  at 
the  rate  of  10  miles  an  hour.  They  all  reach  their  destinatio') 
at  the  same  time.     Wiiat  was  its  distance  and  the  rate  of  liding  'i 

18.  In  a  right-angled  triangle  there  is  given  the  hypothe- 
nuse  =  iif  and  the  area  =  V^]  Und  the  sides. 

19.  Find  two  nundicrs  such  that  their  i)r()duct,  sum,  and 
difference  of  squares  shall  be  eciual  to  each  other. 

20.  Find  two  numbers  whose  product  is  210;  and  if  the 
greater  be  diminished  by  4,  and  the  less  increased  by  3,  the 
l)roduct  of  this  sum  atid  difference  may  be  240. 

21.  There  are  two  numliers  whose  sum  is  74,  and  the  sum 
of  their  S(puirc  roots  is  12.     What  are  the  numbers  ? 

22.  Find  two  numbers  whose  sum  is  72,  and  the  sum  of 
their  cube  roots  G. 

23.  "^riie  sides  of  a  given  rectangle  are  m  and  n.  Find  the 
sides  of  another  which  shall  have  twice  the  perimeter  and  twice 
the  area  of  the  givi'U  one. 

24.  A  certain  number  of  workmen  recpiirc  3  days  to  com- 
plete a  work.  A  number  4  less,  working  3  hours  less  per  day, 
will  do  it  in  0  days.  A  nundjcr  G  greater  than  the  original 
number,  working  (!  hours  less  per  day,  will  com})lete  the  work 
in  4  days.  What  was  the  original  number  of  workmen,  and 
how  long  did  they  work  per  day  ? 

25.  Find  two  numlters  whose  sum  is  18  and  the  sum  of 
their  fourth  powers  1409G. 

Note.  Since  tlie  sum  of  the  two  uunibers  is  18,  it  is  evident  tliat 
the  one  must  bo  as  much  loss  tlian  9  as  the  otlier  is  greater.  'J'lic  ('(jiia- 
tions  will  assume  the  simplest  form  ■when  we  take,  as  the  unknown  quan- 
tity, the  common  amount  by  whicli  the  numbers  ditiVr  from  9, 

26.  Find  two  numbers,  .r  and  y,  such  that 

a?  -\-  if  :  1^  —  if    : :    35  :  19, 
Qcij  =  24. 

27.  Find  two  ininibers  whose  sum  is  14  and  the  sum  of 
their  lifth  powers  1G12U4. 


I     i 

ft! 


ti  1 


interest,  <o 
to  of  iiiler- 
LT,  it  would 
rate  ? 

rate.  Five 
3  miles  ;iu 
stiirti'd  ;it 
(lestinatio-! 
;  of  lidiii;; ':* 

le  liypotlic- 

;,  sum,  iiiid 

and  if  tlio 
d  by  ;},  the 

id  the  sum 
'J 

the  sum  of 

Fiud  the 
;raud  twice 


xys  to  com- 
ss  J)*-'!'  day, 
le  ori^iiuid 

te  the  work 
kmen,  and 

the  sum  of 

rvident  tliat 
'I'Ik'  (Mjiia- 
uiowii  quiin- 
1), 


BOOK    VII. 
PROGRJ':  ss/oxs. 

CHAPTER     I. 
ARITHMETICAL      PROGRESSION. 

2().S.  D(f.  AVhon  wo  liavo  a  Sinics  of  nuiubcrs  oacli 
of  which  is  gr<'ator  or  less  tliaii  tlio  procediiiij:  l)v  a  coii' 
8tant  qnaiitity.  the  series  is  said  to  Ibriu  an  Arithmet- 
ical Progression. 

ExAMpj.i:,     The  series 


7      1  •"»      17 


3'^ 


1 


tliG  sum  of 


etc.  ; 
1 ,     —  3,     etc.  ; 
a  -f  h,     a,     a  —  h,     a  —  )lb,     a  —  oh,    etc., 

are  oacli  in  arithnietieal  ]»rogression,  Ijecause,  in  the  first,  each 
number  is  greater  than  the  preceding  by  5  ;  in  the  second, 
each  is  less  than  the  [)receding  l)y  2;  in  the  tliird,  each  is  le.-^s 
than  tlie  preceding  by  b. 

I)rf.  The  amount  by  wliich  eacli  term  of  an  arith- 
metical j)rogression  is  greater  tliaii  the  preceding  one  is 
called  tlie  Common  Difference. 

Def.  The  Arithmetical  Mean  of  two  quantities  is 
lialf  their  sum. 

All  i\\v  tei'nis  of  an  arithmc^tical  progression  (^xcept 
the  first  and  last  are  called  so  many  arithmetical  means 
between  the  iirst  and  last  as  extrem(_^s. 

Example.  Tlie  four  munbers.  5,  8,  11,1-1,  form  the  four 
arithmetical  means  between  %  and  17. 


\  v\ 


■'  ?  !• 


202 


APJTUMEriCAL    rilOGEESSION. 


% 


EXERCISES. 

1.  Foi'm  loiir  terms   of   tlio   aritliinctical   progression   of 
which  Ih.G  lirst  torin  is  7  and  common  dill'erencc  3. 

2.  Write  the  (Irj-t  seven  terms  of  the  progression  of  which 
the  first  term  is  11  and  tlie  common  ditl'erence  — 3. 

3.  AVrite  five  terms  of  the  progression  of  wliicli  the  first 
term  is  a  —  4:/i  and  the  common  dilference  2?^^. 


^ 


*■  ■ 

1', 

■     ■ 

^i     :  S 

i  '.! 

' 

11 

»l 


Problems  in  Proj^rcssioii. 

209.  Let  us  put 

a,  tlic  first  term  of  a  progression. 
cl,  tlie  common  difference. 
11,  tlie  numher  of  terms. 
/,   the  last  term. 
X,  the  sum  of  all  the  terms. 
The  series  is  then 

a,    a  +  il,    a-\-2(I,  ,...!. 

Any  three  of  the  above  five  quantities  being  given,  the 
other  two  may  be  found. 

Proulhm  I.  Giveiv  the  first  term,  the  cmmiinn  dijfer- 
cncc,  and  tlto  nmnbev  oftcnns,  to  find  the  last  term. 

The  1st  term  is  here  a, 

2d      "         "  a  +  d, 

3d      "        "  a  +  )ld. 

The  coefficient  of  d  is,  in  each  case,  1  less  than  the  number 
of  tlie  term.  Since  this  coefhcient  increases  by  unity  for  every 
term  we  add,  it  must  remain  less  by  unity  than  the  number  of 
the  term.     Hence, 

The.  P^  term  is  a  +  (/  —  1)  d, 
whaicver  be  L     Hence,  when  i  =  7i,  i 

I  =  a -{-  {u-^)d.  (1) 

From  this  equation  we  can  solve  the  further  problems : 

PROHLHM  11.  (liven  the  last  term  1,  the  connnnn,  dif- 
ference (I,  and  ike  naniber  of  terms  it,  to  find  tlie  first 
term. 


f 


^BoawaaaMa 


inmltei' 


t 


f 


Ann  IIMETICAL    PROGRESSION. 


203 


The  solution  is  found  by  solving  (1)  wiLh  respect  to  a, 

l-(n-l)(l  (2) 


"which  gives 


a 


Problem  III.  Gu'cii  the  first  and  last  terms,  a  and  I, 
and  the  nainber  of  terms  n,  to  find  the  common  di/fer- 
ence. 

Solution  Ivom  (1),  d  being  the  unknown  quantity, 


d  = 


n—i 

Problem  IY.     Given  the  first  and  last  teinns  and  the 
common  difference,  to  find  tlie  member  of  terms. 

Solution,  al.^o  from  (1), 

I  —  a  .   ^        I  —  a  -\-  d 


71  r= 


d 


+  1 


d 


(4) 


Problem  V.     To  find  the  sum  of  all  the  terms  of  an 
arithmetical  progression. 

We  have,  by  the  definition  of  2, 
S  =  rt  +  (a  +  rZ)  +  {a  +  M)  -\- {I  —  d)  +  ?, 

the  parentheses  being  used  only  to  distinguish  the  terms. 

Now  let  us  Avrite  the  terms  in  reverse  order.     The  term 
before  the  last  is  I  —  d,  the  second  one  before  it  I  —  2d,  etc. 

We  therefore  have, 

I.  =  I  ^  [l  —  d)  +  {I  —  2d) -\-  {a  -\-  d)  -i-  a. 

Adding  these  two  values  of  2  together,  term  by  term,  we 
find 
21  =  (a  +  l)  +  {a  +  l)  +  {n  +  i)  + +  («  +  /)  +  {a  +  I), 

the  quantity  (n  +  l)  being  written  as  often  as  there  are  terms, 
that  is,  n  times.    Hence, 

21  =  7i{a  -\-  I), 
a  +J 


1  —  n 


(5) 


Remark.    The  expression  — — ,  that  is,  half  the  sum  of 
the  extreme  terms,  is  the  mean  value  of  all  the  terms.     The 


ua  it 


ii! 


204 


A RITIIMETICAL    rROaRESSrOK 


snni  of  the  n  terms  is  therefore  the  same  as  if  cacli  of  tlicm 
luid  this  vahie. 

^lO,  Til  tlie  C(iuati()n  (5)  we  are  supposed  to  know  tlio 
first  and  last  terms  and  the  numher  of  terms.  If  other  (|iuui- 
titics  arc  taken  as  the  known  ones,  we  have  to  snhstitute  for 
some  one  of  the  (piantities  in  (o)  ils  expression  in  one  of  the 
(•(juations  (1),  {'i),  (o),  or  (4).  Sui)})ose,  for  e.\anii)le,  tlmt  we 
iiavc  given  only  the  last  term,  the  eommon  dilferenee,  and  the 
nninl)L'r  of  terms,  (hat  is,  /,  d,  and  n.  We  must  then  in  (o) 
substitute  for  a  its  vahie  in  (:i).     This  will  give, 


n 


/j      11—  1    A  n  {n  —  \) 

y  -  --z-  V  = "'  — I—  ''• 


{'■■) 


EXERCISES, 

In  arithraetietil  i)rogrcssion  tliere  arc 

1.  (liven,  common  diiference,  +  3;  tliird  term  =  10. 
Find  jirst  term.  Afis.    First  term  :=  I, 

2.  (Jivcn  -fth  term  =  I),  common  ditference  =  — c. 
Find  iirst  7  terms,  their  sum  and  product. 

3.  Given  3d  term  =  a  -{-  b,  4th  term  =  a  -{-  :lh. 
Find  first  5  terms. 

4.  (iiven  1st  term  z=  a  —  h,  9th  term  =  i)(i  +  Ti. 
Find  Jid  term  and  common  ilitrerence. 

5.  Given,  sum  of  9  terms  =  108. 

Find  middle  term  and  sum  of  1st  and  tUli  terms. 

6.  Given  nth  term  =  i.'' — '"J//,  ith  term  =  9.r  —  9_y. 
Find  first  7  terms  and  common  ditference. 

7.  Given  1st  term  =  1'^,  HOth  term  =  bo\. 
V\\n\  sum  of  all  ."iO  terms. 

8.  To  lind  the  sum  of  the  first  100  numhcrs,  namely, 

1  +  ;3  +  3 +  99  +  100. 

Here  tlio  first  tenn  n  is  1,  the  lust  tonu  I  100,  and  the  number  of 
terms  100.     Tlio  solution  is  by  Problem  V. 

(J.   Find  the  sum  of  the  Iirst  n  entire  numhcrs,  namely, 

1  +  24-3....  -\-  n. 


AIlITIIMETrCAL    VROOnESSlOX. 


205 


(C) 


i 


I 


10.  Find  tlie  sum  of  tho  first  n  odd  .aimbers,  uamcly, 

1  +  :]  +  5  .  .  .  .  -f  •,'/<  —  1. 
Here  the;  number  of  terms  is  «. 

1 1.  Find  the  sum  of  tho  lir.st  )i  oven  uumbors,  namely, 

^  +  -i  +  <;....  +  -in. 
T2.   In  a  scliool  of  vi  scholars,   th-  hi:j:ho?t  roocivod  lot 
luoi'it  inari\S,  and  each  .sucoeodinix  one  0  Iojjs  than  tho  one  next 
ahovo  him.     How  many  did  the  lowest  scholar  receive?     How 
many  did  tliey  all  receive  V 

13.  Tlie  ilrst  term  of  a  series  is  m,  the  last  term  'hu,  and 
tho  common  diU'erence  d.     \\  hat  is  tho  iiumher  of  terms ':* 

14.  Tlie  first  term  is  Ic,  tho  last  term  l<'/i- — 1,  and  the 
number  of  terms  9.     What  is  the  common  diift.-renco  ? 

15.  The  middle  term  of  a  progression  is  s,  the  number  of 
terms  5,  and  the  common  ditforence  —  //.  AVhat  arc  the  lirst 
and  last  terms  and  the  sum  of  the  5  terms? 

16.  The  sum  of  5  numbers  in  arithmetical  progression  is 

20  and  the  sum  of  their  S([uares  120.     What  are  the  numl)ers? 

NoTK.  In  (juestii)iis  like  this  it  is  better  to  take  the  middle  term  for 
one  01  the  unknown  (juantities.  Tho  oiLer  uukuuwu  tnuuitity  will  bo 
tho  conniioii  ditrereiice. 

17.  Find  a  number  consisting  of  three  digits  in  arithmeti- 
cal progression,  of  which  tho  sum  is  15.  If  the  number  be 
dinnnishod  by  102,  the  digits  will  be  reversed. 

iS.  The  continued  product  of  three  niT'nbors  in  arithmet- 
ical progression  is  640,  and  the  third  is  four  times  tlie  lirst. 
What  are  the  numbers? 

19.  A  travel' or  has  a  journey  (»f  i;]2  miles  to  perf(.rm.    Ho 

goes  27  miles  tho  first  day,  24  the  second,  and  so  on,  travelling 

3  miles  les;  each  day  than  tho  day  before.     In  how  many  days 

will  he  comi)lete  Iho  journey? 

Here  we  luive  liiveii  the  first  term  '27.  the  eomnmn  difTerenoe  —'•),  iind 
the  sum  of  the  terms  V-Vl.  To  solve  this,  we  t:dvi'  e<|uati<in  ^.'ii,  tiiid  sub- 
slituto  for  1  its  value  in  (1).     This  makes  (.li  reduced  to 

a  4-  n  4-  in  —  I)  d  n  in  —  \)  d 

2  2 

2,  (I,  and  (/  are  niven  by  {\\v  problem,  and  n  is  the  unknown  quan- 
tity. Substituting  the  numerical  value  of  the  unknown  (quantities,  tho 
equation  becomes  " 


200 


AlilTILVm'ICAL   rJiOaiiES.iI02i. 


132  =  27)1  —  3 


n  {n  —  1) 

i 

'Z 


Tliis  rfdiirpd  toa  (lamlratic  p(iuation  in  n,  tlio  solution  of  which  {jivos 
two  vahics  of  It.  The  stiuiriit  should  cxphiin  tliis  doiihlo  answer  by 
contiiminj^'-  tlic  progn.'ss-uon  to  11  tcrais,  and  showing  what  the  negative 
ti-rms  indicate. 

20.  Tiiking  tliG  sniiio  question  as  the  last,  only  suppose  the 
tlistance  to  be  140  miles  ii-.stead  of  V.Vi.  Show  that  the  answer 
"will  he  iniMi,Miiary,  and  ex})hiln  this  result. 

21.  A  (lel)tor  owin^-  $100  arrauj^ed  to  pay  25  dollars  the 
first  month,  23  the  second,  and  so  on,  2  dollars  less  each 
montli,  until  his  dcl)t  should  he  discharged,  llow  many  pay- 
ments must  he  make,  and  what  is  the  explanation  of  the  two 
answers  ? 

22.  A  hogshead  holding  135  gallons  has  3  gallons  poured 
into  it  the  first  day,  G  the  second,  and  so  on,  3  gallons  more 
every  day.     IIow  long  l)et'ore  it  will  he  filled  ? 

23.  The  continued  jiroduct  of  5  consecutive  terms  is  12320 
and  their  sum  40.     AV hat  is  the  progression  ? 

24.  Show  that  the  condition  that  three  numhcrs,  jh  (J,  and 
r,  are  in  arithmetical  progression  may  be  expressed  in  the  form 

q  —  r 

25.  In  a  progression  consisting  of  10  terms,  the  sum  of  the 
1st,  3d,  5th,  7th,  and  0th  terms  is  90,  and  the  sum  of  the  re- 
maining terms  is  110.     What  is  the  progression  ? 

26.  In  a  progression  of  an  odd  numljcr  of  terms  there  is 
given  the  sum  of  the  odd  terms  (the  first,  third,  fifth,  etc.) 
{iiid   the  sum  of  the  even  terms   (the  second,  fourth,  etc.). 
Show  that  we  can  find  the  middle   term  tind  the  number  of 
terms,  but  not  the  common  diil'erence. 

27.  In  a  jirogression  of  an  even  number  of  terms  is  given 
the  sum  of  the  even  terms  —  105,  the  sum  of  the  odd  terms  = 
110.  and  the  excess  of  the  last  term  over  the  first  =  2G.  AVhat 
is  the  progression  ? 

28.  Ciiven  a  and  /,  the  first  and  last  terms,  it  is  rerpiired  to 
insert  i  arithmetical  means  between  them.  Find  the  expres- 
sion for  the  i  terms  required. 


I 


UEOMin'uit'AL  PUoaniJ.'^sioN. 


207 


ich  gives 

iifswcr  by 

negative 

pose  the 
.'  iinswcr 

lars  tlic 
.'ss  each 
my  piiy- 
the  two 

3  poured 
lis  more 

is  12;3^0 

),  q,  and 
he  form 


of  the 
the  re- 
there  is 
li,  etc.), 
|i,   etc.). 
inbcr  of 

|is  given 
■rms  = 
AVhafc 

hired  to 
lexpres- 


CHAPTER     II. 
GEOMETRICAL     PROGRESSION. 

'ill.  Dcf.  A  Geometrical  Progression  consists  of 
11  series  of  terms  of  wliich  each  is  formed  by  miiltii)ly- 
iiig  the  term  preceding  by  n  (jonstciiit  factor. 

An  arithmetical  progression  is  fo'/nied  by  continual  a(kU- 
tioii  or  subtraction;  a  geoniutrical  progression  by  repeated 
niuUij>licatioii  or  division. 

Dcf.  Tlio  factor  by  which  each  tei-m  is  multiplied 
to  form  the  next  one  is  called  the  Common  Ratio. 

The  common  ratio  is  analogous  to  the  common  (lill'erence 
in  an  arithmetical  })rogression. 

In  other  respects  the  same  definitions  ajjply  to  both. 

EXAMPLES. 

9,     G,     LS     iA,     etc., 
is  Ji  progression  in  Avhich  the  first  term  is  2  and  the  comnion 
ratio  3. 

^'     ^'     2'     4'     8' 
is  a  progression  in  wliich  the  ratio  is  -• 


etc., 


+  o 


0,     +  1:1,     —  24,     etc., 


is  a  progression  in  which  the  ratio  is  —  2. 

Sort).  A  ])r()i:rcHsi{)n  Vvhi^  thr  scm-oikI  one  ii1)()Vf'.  fornied  by  dividiiiii" 
each  term  hy  tlic  Hiuiie  divisor  to  obtiiiii  the  next  tt'rni,  is  included  in  the 
gcnural  <lelinition,  bpcauso  dividing  by  any  number  \»  tlio  «unu'  as  inulti- 
jjlying  i\v  the  reriprocal.  (icometrical  progressions  may  therefore  be  , 
divided  into  two  classes,  inrrcusing  and  decreasing.  In  the  increasing 
progression  the  common  ratio  is  greater  than  1  and  the  teuns  go  on  in- 
creasing ;  in  a  diminishing  progression  the  ratio  is  less  than  unity  and 
the  terms  go  on  diminishing. 

Rem.     Ill  a  i^rogression  in  which  the  ratio  is  negative,  the 
terms  will  be  alternately  positive  and  negative. 


\U 


208 


O EOMETRW.  1  /.    7' ROCrliJi'SSlOI^. 


Drf.     A  Geometrical  Mean  between  two  quantities 
is  the  scxuare  root  of  tlieir  i)ruduct. 

EXERCISES. 

Form  five  terms  of  each  of  the  i'olluwiiig  gcomctriciil  pro- 
grurfsiuiis : 

1.  First  term,  1  ;  common  ratio,  5. 

2.  First  term,  7;  common  ralio,  — '). 

3.  i"'irst  term,  1  ;  common  ratio,  —  1. 

4.  First  term,  ^  ;  common  ratio,  '  • 

4  .      1 

5.  First  term,  .  ;  common  ratio,  ;- 


I 


i  ] 


Problems  of*  Geoiuelrical  I*ro.i»r(vssioii. 

2\2,  In  a  <2:eometrical  progression,  as  in  an  arithmetical 
one,  thei'c  are  live  ((uaiitities,  any  thiee  of  wliieh  (U'lermiiiu 
the  progression,  and  enable  the  other  two  to  be  funnd.     'I'hey 

are 

a,    tlie  first  term. 

r,  the  common  I'atio. 

11,  the  nnmljcr  of  terms. 

/,  tlio  last  term. 

1,  the  sum  of  the  n  terms. 

The  general   expression   for   the   geometrical   progression 

■will  be 

a,    ar,     ar%    a)'\    etc., 

because  each  of  these  terms  is  formed  by  multiplying  tlie  pre- 
ceding one  by  r. 

The  same  [)roblems  present  themselves  in  the  two  progres- 
sions.    Those  for  the  aeuinetrical  one  are  as  follows: 


Si^ 


PuoiiLE.M  I.     Oirrii  the  J/'rsf  fcrin,  the  commoii  ratio, 
and  the  iiuiiil)ei'  oj'  fcrn/s,  to  ]iiid  the  last  tcrnb. 

The  progression  will  ha 

a,     ar,     ar^,     etc. 

"Wo  see  that  the  exponent  of  /•  is  less  by  1  than  the  numl)er 
of  the  term,  and  since  it  increases  by  1  for  each  term  added,  it 


J 


gi:omi:tr[c.  i  /.  rnoGiiEssioy. 


200 


pro- 


must  romiiin   less   by  1,  liow  many  terms  so  over  ^vc  take. 
Ik'iicc  tliu  u"''  term  is 

/  =  ar"-i.  (1 ) 

PiioBLEM  IT.     Civni  tha  last  tern} ,  ihe  cnvnnon  ralio, 
and  Ihe  nnnibcr  of  terms,  to  Jiiid  the  Jirst  term. 

The  .solution  is  I'uuud  by  dividing  both  members  of  (1)  by 

r"~',  wiiich  gives 

7 


a  = 


y"-i 


PuoriLF.M  III.     Ctiveii  the  /Irst  term,  the  last  term,  and 
the  nuniher  of  terms,  to  Jind  the  eomnioii  ratio. 

From  (1)  we  tlnd  r^'^  =  - 

Extracting  the  {n  —  1)"'  root  of  each  mcmljcr,  we  have 


r 


= cr  • 


[The  solution  of  Problem  IV  requires  us  to  find  n  from 
equation  (1),  and  belongs  to  a  higher  department  of  Algebra.J 

Proulf.m  V.     To  find  the  sum  of  cdl  n  terms  of  a  ^eo- 
mctriecd  j)roj^rcssioii. 

AVc  have     E  =  a  +  ar  +  af-  +  etc.  -f-  ar^^~'^. 
Multii)ly  both  sides  of  this  equation  by  r.    We  then  have 
r'E  =  ar  +  ar'^  +  ar^  +  etc +  ar^K 

Now  subtract  the  first  of  these  equations  from  the  second. 
It  is  evident  that,  in  the  second  e(juati<)n,  each  term  of  the 
second  memljcr  is  Cf(ual  to  the  term  of  the  second  member  of 
the  first  equation  which  is  one  place  farther  to   the   rio-ht. 
Hence,  when  we  subtract,  all  the  terms  will  cancel  eiich  other 
except  the  first  of  the  first  equation  and  the  last  of  the  second. 
Illustuatiox.     The  following  is  a  case  in  which  r<  =  2,  r  =  3,  w  =  G : 
2  :=  2  +  G  +  18  +  O4+1G2  +  480. 
32  -  G  + 18  +  54  +  1G2  +  480  + 1458. 
Subtracting,    32-2  =  1458  -  2  :=  145G, 

or    22  =  1450,    and     2  =  738. 
14 


! 


,      ; 


210  GICOMETIUCAL    PUOaRKSSrON. 

lii'turniii;,'  to  the  cfciioral  pr()l)loni,  wc  have 

(;•  -  1)  i  =  (()'''  —  a  =  (I  (/•»  —  1) ; 

Avhcncc, 


r'i  _  1  1  _  r» 

/•  —  1  1  —  ;• 


(1) 


It  will  1)C  most  convcnicMit  to  iiso  the  first  form  when  r  >  1, 
and  the  second  wlien  r  <  1. 

liy  this  forinuhi  we  are  enabled  to  rompute  the  sum  of  tlio 
terms  of  a  j^eometrica!  i)r(>gression  without  actually  foruiiii*^ 
all  tlie  terms  and  aihliiiir  lliein. 


t 


EXERCISES 


3 


I    it 


1.  r.iven  3d  term  =  9,  common  ratio  =  ^. 
rind  lirst  5  terms. 

32  2 

2.  Cliven  5tl»  term  =  ' '^,  common  ratio  =  — -• 

Find  iirst  5  terms. 

3.  f liven  -"ith  term  =  xhp,  1st  term  =  ij^. 
Find  common  ratio. 

4.  {jiiven  1st  term  =  1,  4th  term  =  cA 
Find  common  ratio  and  lirst  3  terms. 

5.  Given  2d  term  =  vi,  common  ratio  =  —  m. 
Find  first  4  te.ms. 

6.  A  farrier  havings  told  a  coachman  that  he  wonld  charge 
him  13  for  slioeing  liis  horse,  the  latter  objected  to  the  price. 
Tlie  farrier  then  olTered  to  take  1  cent  for  the  lirst  nail,  2  for 
the  scecmfl,  4  for  the  third,  and  so  on,  doubling  the  amount 
for  each  nail,  which  offer  tlie  coachman  accepted.  I'liere  were 
32  nails.  Find  how  much  the  coaclunan  had  to  pay  for  the 
last  nail,  and  hoAV  mucli  in  all.     (Compare  §  1G8,  IiEM.) 

7.  Find  the  sum  of  11  terms  of  the  series 

2  +  <3  +  18  -\-  etc., 

ill  Avhich  the  first  term  is  2  and  the  common  ratio  3. 

8.  If  the  common  ratio  of  a  progression  is  r,  what  will  be 
the  common  ratio  of  the  progression  formed  by  taking 

I.  Every  alternate  term  of  the  given  progression? 
II.   Every  ?i^^''  term  ? 


VI 


llG 


I 


OEOMETIUrAL    PUnnUESSlOK 


211 


9.  The  snnio  tliiniij  l)('iii;j^  sujtprjscd,  wlitit  will  l>c  (ho  com- 
iiKni  ratio  of  tho  |)i'<»<riv.ssioii  of  wliich  every  iillcrimte  term  id 
e<|iiiil  to  every  third  tiriii  of  the  ^iveu  pro^Mvssioii  '•' 

10.  Show  that  if,  in  u  geometrieal  i)r»»oT(.'ssioii,  each  term 
be  nclded  to  or  .siil)trac'ted  from  that  next  following,  the  jsunid 
or  remainders  will  form  a  ^'eomotrieal  proj^ressiou. 

11.  Show  that  if  tho  arithmetical  and  geometrical  means 
of  two  quantities  be  given,  the  (juantlties  themselves  nniy  be 
found,  ami  give  tho  expressions  for  them. 

12.  Tho  sum  of  the  first  and  fourth  terms  of  a  jn'ogression 
is  to  the  sum  of  the  second  and  third  as  'li  :  5.  What  is  tho 
common  ratio? 

13.  Express  tho  continued  product  of  all  tho  terms  of  a 
geometrical  progression  iu  terms  of  «,  r,  and  w? 

IJiiiit  <)f  tlic  Sum  of  a  l*r<)j»rossi<)ii. 

21.3.  Theorem.  If  the  coniinon  ratio  in  a  gooiiiotri- 
cal  i)r()g'rossion  is  less  than  unity  (more  exactly,  if  it  is 
contained  between  the  limits  —1  and  +1),  then  there 
will  be  a  certain  quantity  which  the  sum  of  all  tin; 
terms  can  never  exceed,  no  matter  liuw  many  terms  we 
take. 

For  example,  tho  sum  of  the  })rogrcssion 

11,1 

^4-;^  +  y  +  etc., 

in  Avliich  the  common  ratio  is  -,  can  never  amount  to  1,  no 

matter  how  many  terms  we  take.  To  show  this,  suppose  that 
one  person  owed  another  a  dollar,  and  proceeded  to  pay  him  a 
series  of  fractions  of  a  dollar  in  cfoometrical  proirression. 
naniel 


^ly 


2' 


1 

10 ' 


etc. 


"When  ho  paid  him  tho  -  ho  would  still  owe  another   ., 
when  ho  paid  tho  ~.  he  would  still  owe  another  -  ,  and  so  on. 


I! 


11 


;i 


212 


(}Eo\fi:rni(\  i  l  r/innn/:ssioy. 


That  is,  at  cvory  piiymcnt  lio  would  disclmr^^o  one-li;iIf  llio  ro- 
inainiii';  (Iclit.  Sow  tlifiv  aro  two  iJi'opoHitioiis  lo  Ik-  iiiiilrr- 
Htt»(»tl  ill  rctVri'tK'o  to  (liis  suhjirt. 

I.  'I'/tr  cufirc  dthb  viiii  lurcrhc  (lisrjtfirjjcil  by  such 
jnnjniciilx. 

For,  since  the  ildit  is  halved  \\i  every  ptiynioiit,  if  there  was 
any  |)ayineiit  which  (lischur^'ed  the  whole  reiiiaiiiin<,'  dehl,  tho 
half  of  Ji  tiiiji^'  wmild  hi-  equal  to  the  whole  of  it,  which  is 
irn])ossiI)le. 

H.  77//!  ilchi  run  hr  red  nerd  hi  hue  (inij  assi '^liable 
limit  hij  rontinin'n^  to  fuitj  hutf  of  it. 

For,  liowever  small  the  dcht  may  he  made,  arolher  pay- 
ment will  make  it  smaller  hy  one-half;  hence  thcro  is  no 
smallest  amount  helow  which  it  cannot  he  reduced. 

Thcso  two  propositions,  whicli  fiocm  to  op))i)sc  cncli  otlicr,  hold  tlin 
truth  l)otween  tlicm,  us  It  were.  Tlioy  cotixtniilly  ciitcT  into  tlio  lii/iht-r 
iniithciuaticH,  mid  should  bo  wt-Il  understood.  Wt!  therefore  present 
another  illustration  oi  the  same  subject. 


I 


B 


A 


1 


1 

10 


Suppose  x\B  to  ho  a  line  of  given  length.  Lot  us  go  one- 
half  the  distance  from  A  to  B  at  one  ste[),  one-fourth  at  tho 
second,  one-eighth  at  the  third,  etc.  It  is  evident  that,  at  each 
step,  we  go  half  the  distance  which  remains,  lli'nce  the  two 
princiidos  just  cited  ap})ly  to  this  case.     That  is, 

1.  Wc  can  never  reach  B  by  a  series  of  such  steps,  hecausc 
Avc  shall  always  have  a  distance  equal  to  the  last  step  left. 

2.  But  we  can  come  as  near  B  as  we  ]dease,  because  every 
sto])  carries  us  over  half  the  remaining  distance. 

Tliis  residt  is  often  expri'ssed  by  siiyin'j;  that  we  .should  rearh  B  by 
taking  an  inlinite  number  of  steps.  'I'Ui.-i  is  a  convenient  form  of  expres- 
Bion,  and  we  may  sometimes  use  it,  but  it  is  not  lofi^ically  exact,  because 
no  conc'ivable  nuinljer  can  be  really  iulinitc!.  Tho  assumption  that  in- 
finity is  an  alifcl.niic  (|uaiitity  often  leads  to  ambiguities  and  tlitliculties 
iu  the  a])i)licallou  of  nuitheuiiUics. 


G Ko.yr/:  run:,  i  a  rnoa nh'ssroN. 


2i:] 


•HUSO 


I 


Drf.  Tlio  Limit  of  tlio  sum  i  of  si  ^oomrtiicMl 
progression  is  ji  (juaiitity  wiiicii  l  may  {ipproncii  so 
tiial  its  (liirciciicc  shall  he  less  than  any  (jiiaiitily  u<' 
choose  to  assign,  l)iit  whieli  1  ran  never  reach. 


E  X  A  M  r  L  E  S 

I.  T^iiity  is  tlic  liiiiil  <•!'  tjic  -tim 
1.1.1         1 


etc. 


5J  ^  4  ^  s   '    k; 

2.  Tlio  point  B  in  llic  iircc'diiit;  lii^urc  is  (lie  liinil  of  all 
the  sh'ps  that  can  he  tukon  in  the  inauiici'  (lr.st'rihc'<l. 

Tlic  followin;;  ])rin('iple  will  cnahk'  us  to  liiul  the  limit  (»!' 
the  sum  of  !i  progression  : 

'ill.  Pri/ic//»7('.  ir  /'  <  1,  the  ])o\ver  /•"  can  he  made 
as  small  as  we  ]»lease  \)y  increasing  the  value  oi'v^,  but 
can  lun'er  bo  iiuuh^  ec^ual  to  0. 

►Suppose,  for  instiince,  that 


Then    evei-y    lime   we    miiltijily  hy   r   we  iliminlsh  r"  l>y 
-   of  its  former  value;  that  is. 


,.3  ^  •;  ,2 

4 


yi  _       r'i 


/•*, 


r*  =  -  /'^  =  r^  —    r'\ 


etc. 


etc. 


etc. 


Now  let  us  again  take  the  expression  for  the   sum  of  a 


series  of  n  terms,  namelyj 


1  =  a 


1  —V 


1  —  r 


which  wc  may  put  into  the  form 


2  = 


a 


a 


1  —  /•       1  - 


HA 


214 


oeomethtcal  rnoanESSioN. 


Hi 


If  r  is  loss  tlmn  unity,  avc  can,  by  tlie  principle  just  cited, 
make  tlie  ([uantily  r'*  as  small  as  we  ]»lease  by  incivasini:^  n 
indelinitely.     From  tliis  it  iollows  that  we  can  alsu  make  the 


term 


a 


r"'  as  small  as  Ave  jikase. 


Proof.     Let  us  put,  for  brevity. 


k  = 


a 


so  that  the  term  under  consideration  is 

kr'K 

If  we  cannot  make  Zv'^  as  small  as  we  please.  sn])pose  ,s  to 
be  its  smallest  possible  value.     Let  us  divide  a  by  /c,  and  put 


/  = 


s 


No  matter  how  small  s  may  be,  and  how  large  Jc  may  be, 
V,  or  /,  will  always  be  greater  than  zero,  llcnce,  by  the  pre- 
ceding jirineiiile,  we  can  find  a  value  of  }i  so  great  tluit  r'' 
shall  be  less  than  /.     That  is, 


< 


> 


11  I 

J. 


Multiplying  both  sides  of  this  inequality  ])y  i; 

That  is,  liowevov  small  we  take  s,  we  can  take  n  so  large 
that  I'r''^  shall  be  less  than  ^•,  and  therefore  s  cannot  be  the 
smallest  valu3. 


S 


mce 


2   =: 


a, 


—  Jcr\ 


and  since  we  can  make  7cV"  as  small  as  we  please,  it  follows 


that 


a 


Limit  of  2  =  ^ 


Th 


IS  IS 


o'Unetimes  expressed  by  saying  that  when  r  <  Ij 


a 


+  ar  +  ur-  +  (O'^  +  etc.,  ad  infinitum  = 


a 


1  —r 


and  this  is  a  convenient  form  of  expresaion,  which  will  not  lead 
us  into  error  in  this  ease. 


GEO  METRIC  A  L    PROGRESSION. 


215 


the 


[lows 


lead 


EXERCISES. 

Ilaviug  given  the  progression 

1111 

2+4  +  8  +  10  +  ^^^" 

of  Avliich  the  limit  is  1,  find  how  nianv  terms  wo  nnist  talvo  in 
order  that  tlie  sum  may  diil'er  from  1  by  less  than  the  follow- 
ing (juantities,  namely: 

Firstly,  .001 ;  secondly,  .000  001 ;  thirdly,  .000  000  001. 

To  do  this,  we  ninst   find  what  power  of  ^^  will  be  less  than  .001, 
what  power  lesrf  than  .000  UOl,  etc. 

AVhat  are  the  limits  of  the  snms  of  the  following  series: 

II  1  ,  7  .    .^     .. 

I.     3  +  33  +  33  +  ^'^^•'  ^'"  injinitum. 

2       4        S 

^  +  fj  +  -^^  +  etc.,  ad  iufinUum. 

III  ,  ;    .      .      ., 

o  ~  f,3  +  03  ~  ^^^'■>  ^^^  infinitum. 

4       42       43 

fj  4-  jj2  +  (]3  +  ^^^'^  ^''^  infinitum. 


2. 

3- 
4- 

5- 
6. 

7- 


T-T-T   + 


+ 


1+  ^>   '   (1  -H  Z»)2  '    (1  +  t>f 
a  a  a 


+  etc.,  (lA  i)ifuiitum. 
—  etc.,  ad  luftnituni. 


1       {h-  1)2  '    {b  —  1)3 

etc.,  ad  infinitum. 


2        12         1 

ni        iii^       iir       i/r 


8.  What  is  that  progression  of  which  the  first  term  is  12 
and  the  limit  of  the  snm  8. 


B 

I 


9,  On  the  line  AB  a  man  starts  from  A  and  goes  to  the 
point  c,  half  way  to  B ;  *hen  he  re- 
turns to  d,  half  "Way  back  to  A  ;  then  ,  ,  ," 
tiiT'is  again  and  goes  half  way  to  c, 
then  back  half  way  to  d,  and  so  on,  going  at  each  turn  half 
way  to  the  point  from  which  he  last  set  out.  To  what  point 
on  the  line  will  he  continually  approach  ? 


I;    , 
I   ,! 


I-!  •*' 


f  I 


'u; 


1/^ 


ri( 


i    i 


210  COMPOUND  interp:st. 

21/5.  As  ail  iiitcro.yfiiip:  :ip])lir;if  ion  of  t.lio  preceding  theory, 
we  may  examine  the  problem  of  liiulinc^  the  vakie  of  a  circu- 
]atiii;2^  (leeiinal.  Such  a  (Kcimal  i>  always  e<[nal  to  a  vulgar 
fraetioii,  wliieli  is  ()l)taiiic(l  as  in  the  following  fxamjjles: 

1.  What  is  the  vahie  of  the  deeinuil 

,'i  i')  iO  i     .    .    .    .    . 

W(>  find  tlio  fig-urf's  wliicli  form  tlic  jioriod  to  bo  37.  Dividing  the 
dociinal  into  periods  of  tln.'sc  fi^-iircs,  it.s  viiluc  in 

O'V  07  oi»y 

-   4-    ~        -I-  -— '      4-      t 
100  "^  100^  "^  1003  ^~ 

=  ^^^(17^0  +  1^30^  +  1005  + ''4 

Tlio  quantity  in  tlio  parcntlicsis  is  a  goonictrical  pron^ropsinn,  in  which 
a  =         ,  r  =       ^  •   The  liiuit  of  its  sum  is  tlierefore  -  ^^  •    Thercfcji-c  thf 

value  of  tlio  decimal  is     ^^  • 

Tliis  result  can  bo  proved  by  changing  this  vulgar  fraction  to  a 
decimal. 

2.  In  the  case  of  a  decimal  which  has  one  or  more  figures 
hcfore  the  period  commences,  wc  cut  these  fignres  olf,  and 
find  the  value  of  them  and  of  the  circulating  part  separately. 
Thus, 

503C3  etc.  =  ^  +  j^,,,^  +  J-—  +  etc. 

_   5         0:5    /  1_         1  \ 

~  10  "^  loiJo  I    "^  100  "^  roo2  "^      7 

_   5  03     100  _   5         03^  _  558  _  31 

~  10  "^  loOO'W  ~  10  "*"  D'JO  ~  'J'JO  ""  55" 

EXERCISES. 

To  what  vulgar  fractious  arc  the  following  circulating  deci- 
mals eijual : 

I.     .111111 ?  2.  ,2'2-22 ? 

3.  .9!»0i) ?  4.  .00!)'.);  t ? 

5.  .454545  .  .  .  .  ?  6.  .2454545  .  .  .  .  ? 

7.  .108108 ?  8.  72454545 ? 


11 
ai 
f]' 


COMPOUND   INTEREST. 


217 


hcory, 
circu- 
vulgar 


(ling  tlic 


,  in  wliH'li 
rc'fcnx'  tilt' 

ctioii  to  a 

;c  figuves 

olT,  and 

>purately. 


•) 

DO 


31 

55 


tmj,^  tlcci- 


Coiiipouiid  Interest. 

21 G.  When  one  loans  or  invests  money,  collects  the  inter- 
est at  stated  intervals,  and  again  loans  or  invests  this  interest, 
and  so  on,  he  gains  compound  interest. 

Compound  interest  can  always  be  gained  by  one  who  con- 
stantly invests  all  his  income  derived  from  interest,  provided 
that  he  always  collects  the  interest  when  due,  and  is  al)le  to 
loan  or  invest  it  at  the  same  rate  as  he  loaned  his  principa'. 

Problem  I.  To  fiul  the  mnonnt  of  2P  dollars  for  ii 
years,  at  c  per  cent,  conipouud  interest. 

Solution.     At  the  end  of  one  year  the  interest  will  be 

-^-,  which  added  to  the  principal  will  make  ;m  t  +  ^  >   )• 
100'  -^  ^  M         100/ 

c 

If  we  put         p  =  ---  =  the  rate  of  annual  gain, 

the  amount  at  the  end  of  the  year  will  be  p{i  -\-  p). 

Kow  suppose  this  whole  amount  is  put  out  for  another 
year  at  the  same  rate.  The  interest  will  be  _^>  (1  +  p)  p,  which 
added  to  the  new  principal  p  (1  +  p)  will  make  p  (1  +  py. 

It  is  evident  that,  in  general,  sujiposing  the  whole  sum 
kept  at  interest,  the  total  amount  of  the  investment  will  be 
multiplied  by  1  +  p  each  year.  Hence  the  amount  at  the  ends 
of  successive  years  will  be 

p{l  +  p),    p{l  +  py,    p{l+pf,    etc. 

At  the  end  of  ?i  3'ears  the  amount  will  be 

P  (1  +  pY- 

Problem  II.  t4  person  puts  out  p  dollars  every  year, 
letting  the  whole  sum  constantly  accumulate  at  com- 
pound interest.  IVhat  will  the  amoufit  he  at  the  end  of 
11  years? 

Solution.  The  first  investment  will  have  been  out  at 
interest  71  years,  the  second  w  —  1  years,  the  third  ?i  —  2  years, 
and  so  on  to  the  w'*,  which  will  have  been  out  1  vear.  Hence, 
from  the  last  formula,  the  amounts  will  lie : 


J 


218 


COMPOUND    INTEREST. 


Amount  of  1st  payment,    p{i  +  p)'^ 


i\ 


(( 


it 


it 


"  2d 

(( 

p{i  +  pY~\ 

"  ad 

a 

j){lJ^p)n-\ 

"  4th 

a 

p  (1  +  p)"-3 

"   oth 

a 

p{i  +  pY-\ 

etc. 

etc. 

The  sum  of  the  amounts  is : 

p{l+p)  ^  p{\+pf  j^  p{i+pY  ^  . .  .  .  pil^pY, 

This  is  a  geometrical  progression,  of  which  the  first  term  is 
p  (1+p),  tlie  common  ratio  1+p,  and  the  number  of  terms  )i. 
So  in  the  formuhi  (4),  §  21'^,  we  put  p  (1  -f  p)  for  a,  l-\-p  for 
r,  and  thus  find, 

^  ^     ■  '  '    1  -{-  p  —  I  ^  p 

EXERCISES. 

1.  A  man  insures  his  hfe  for  $5000  at  the  age  of  30,  pays 
for  his  insurance  a  premium  of  80  dolhirs  a  year  for  32  years, 
and  dies  at  the  age  of  02,  immediately  before  the  33d  payment 
would  have  been  due.  If  the  com])any  gains  4  per  cent,  inter- 
est on  all  its  money,  how  much  does  it  gain  or  lose  by  the 
insurance  ? 

Note.  Computations  of  this  class  can  be  made  with  great  facility  by 
the  aid  of  a  table  of  logarithms. 

2.  What  is  the  present  \alue  of  a  dollarcdue  u  years  hence, 
interest  being  reckoned  at  c  per  cent.  ? 

Note.     If  p  be  the  present  value,  Problem  I  gives  the  equation, 


K^  +  ioo)"='^- 


3.  What  is  the  present  value  of  3  annual  payments,  of  a 
dollars  each,  to  be  made  in  one,  two,  and  three  years,  interest 
being  reckoned  at  5  per  cent.  ? 

4.  What  is  the  present  value  of  w  annual  payments,  of  a 
dollars  each,  the  first  being  due  in  one  year,  if  the  rate  of  in- 
terest is  c  per  cent.  ?  What  would  it  be  if  the  first  payment 
were  due  immediately  ? 


I 


in- 


p)n. 

St  term  is 

:  terms  7i. 

1  +  P  for 

-■ 

-HP). 

f  30,  pays                | 

■  32  years, 
L  payment 
Lnit.  inter- 
ise  by  tlie 

t  facility  by 

ars  hence, 

nation, 


SECOND  PART. 


ADVANCED    COURSE. 


lents,  of  a 
•s,  interest 

lents,  of  a 
rate  of  in- 
t  payment 


it 


• 


'I 


■—•"—• 


BOOK    VIII. 

RELATIONS   BETWEEN  ALGEBRAIC 

QUANTITIES. 


Of  Algebraic  Functions. 

31T.  Def.  AVlion  ono  quantity  depends  npon  an- 
other in  sucli  a  way  that  a  change  in  the  value  of  tlit; 
one  i)i'oduces  a  cliange  in  the  value  of  the  other,  the 
latter  is  called  a  Function  of  the  former. 

This  is  a  more  general  definition  of  the  word  "  function  "  than  that 
given  in  §  49. 

Examples.  The  time  required  to  perform  a  journey  is  a 
function  of  the  distance  because,  other  things  being  equal,  it 
varies  with  the  distance. 

The  cost  of  a  package  of  tea  is  a  function  of  its  wciglit,  be- 
cause tlie  greater  the  weight  the  greater  the  cost. 

An  algebraic  expression  containing  any  symbol  is  a  func- 
tion of  that  symbol,  because  by  giving  different  values  to  the 
symbol  we  shall  obtain  different  values  for  the  expression. 

Dcf.  An  Algebraic  Function  is  one  in  which  the 
relations  of  the  quantities  is  expressed  by  means  of  an 
algebraic  equation. 

ExAMPi  E.  If  in  a  journey  wo  call  t  the  time,  s  the  average 
speed,  and  d  the  distance  to  be  travelled,  the  relation  between 
these  quantities  may  be  expressed  by  the  equation, 

d  =  si. 

4.ny  one  of  these  quantities  is  a  function  of  the  other  two, 
defined  by  means  of  this  equation. 

An  algebraic  function  generally  contains  more  than  one 


'M 


bi:' 


222 


FUNCTIONS. 


letter,  and  therefore  depends  upon  several  quantities.  But  we 
may  consider  it  a  function  of  any  one  of  these  quantities,  se- 
lected at  i)k'asure,  by  su})])ositifif  all  the  other  (luantities  to 
remain  constant  and  only  this  one  to  vary.  For  example,  the 
time  required  for  a  train  to  run  between  two  points  is  a  func- 
tion not  only  of  their  distance  a])art,  but  of  the  s})eed  of  the 
train.  The  si)eed  being  sui)i)osed  constant,  the  time  will  bo 
greater  the  greater  the  distance.  The  distance  being  Constant, 
the  time  will  be  greater  tlu^  less  the  speed. 

Def.  The  quantities  between  whicli  tlie  relation  ex- 
pressed by  a  function  exists  are  called  Variables. 

This  term  is  used  because  sucli  quantities  may  vary  in  value,  as  in 
the  preceding  examples. 

Def.  An  Independent  Variable  is  one  to  which  ^^  e 
may  assign  values  at  pleasure. 

The  function  is  a  dependent  variable,  the  value  of  which  is 
determined  by  the  value  assigned  to  the  inde})endent  variables 

Drf.  A  Constant  is  a  quantity  which  we  suppose 
not  to  vary. 

Rem.  This  division  of  quantities  into  constant  and  varia- 
ble is  merely  a  supposed,  not  a  real  one  ;  we  can,  in  an  algebraic 
expression,  suppose  any  quantities  we  please  to  remain  constant 
and  any  we  please  to  vary.  The  former  are  then,  for  the  time 
being,  constants,  and  the  latter  variables. 

Illustratiox.    If  we  put 

d,  the  distance  from  New  York  to  Chicago  ; 
s,  the  average  speed  of  a  train  between  the  two  cities ; 
t,  the  time  required  for  the  train  to  perform  the  jour- 
ney, 

then,  if  a  manager  computes  the  different  values  of  the  time  t 
corresponding  to  all  values  of  the  speed  .<?,  he  regards  d  as  a. 
constant,  s  as  an  independent  variable,  and  t  as  a  function  ofs. 
If  he  computes  how  fast  the  train  must  run  to  perform  the 
journey  in  different  given  times,  he  regards  t  as  the  independ- 
ent variable,  and  s  as  a  function  of  t. 


4^ 


I 


FUNCTIONS. 


223 


cities ; 

e  jour- 
time  t 
d  as  a 
m  of.s. 
rm  til  6 

lepend- 


I 


I 


Wlicn  we  liave  any  cciiiation  between  two  variiil>les,  we 
may  re<jfard  eitlier  of  them  as  an  independent  variable  and  the 
other  as  a  function. 

Example.    From  the  equation 

ax  -^lij  =  c, 


we  derive 


X 


II 


a   ^  a' 

(fX         c 


in  one  of  whicli  x  is  expressed  as  a  function  of?/,  and  in  tlie 
otiier  y  as  a  function  of  .v. 

218.  Names  are  ^nven  to  particular  classes  of  functions, 
among  which  the  following  are  the  most  common. 

1.  Def.  A  Linear  Function  of  several  variables  is 
one  in  wliicli  eacli  term  contains  one  of  the  variables, 
and  one  only,  as  a  sinij)le  factor. 

Example.    The  expression 

Ax  +  By  +  Cz 

is  a  linear  function  of  .'r,  //,  and  z,  when  A,  B,  and  Care  quan- 
tities which  do  not  contain  these  variables. 

A  linear  function  dill'ers  from  m  function  of  the  first  degree 
(§  iyl)  in  having  no  term  not  multiplied  by  one  of  the  varia- 
bles.    For  example,  the  expression 

Ax  +  By^-C 

is  a  function  of  x  and  y  of  the  first  degree,  but  not  a  linear 
function. 

The  fundamental  property  of  a  linear  function  is  this: 

//  all  the  variables  he  viuJtif)Jie(l  hy  a  coimnoii  fac- 
tor, the  function  irill  be  Dinlti plied  bij  the  same  factor. 

Proof.  Let  Ax  -\-  By  -{-  Cz  be  the  linear  function,  and  r 
the  factor.  ]\rultii)lying  each  of  the  variables  x,  y,  and  z  by 
this  factor,  the  function  will  become 

Arx  +  Bry  -\-  Crz, 
which  is  equal  to         r  {Ax  +  By  -j-  Cz). 


111-, 
.(  .tit I' 


224 


FUNCTIONS. 


i 


Moreover,  ff  linear  fuuctloii  is  the  only  one  which  pos- 
sesses this  properii/. 

2.  D(f.  A  Homogeneous  Function  of  several  va- 
rial)l('s  is  one  in  which  each  tenii  is  of  the  same  degree 
ill  the  variables.     (Compare  §  52.) 

Example.  Tlio  ('X])rossion  aa^-\-h''^//  +  r>/-z-\-(fz^  is  homo- 
geneous and  of  the  (hh'd  de^^rco  in  tlie  variables  x,  i/,  and  z. 

Rem.  a  Hnear  fiiuetiun  is  a  homogeneous  function  of  the 
first  degree. 

FU.VDAMENTAL   PROPERTY    OF   IIOMOOENEGrS  FUNCTIOXS. 

If  all  tlie  variables  he  itutlti plied  hij  a,  coninioii  faetor, 
any  honio^cueoiis  function'  of  the  n*'*  (h'<^ree  iib  those  I'd- 
riahles  icill  he  itiultipUed  by  the  ii^'*^  power  of  that  factor. 

Froof.  If  wc  take  a  homogeneous  function  and  put  rx  for 
X,  ry  for  y,  rz  for  z,  etc.,  then,  because  each  term  contains  .r, 
y,  or  z,  etc.,  n  times  in  all  as  a  factor,  it  will  contain  r  n  times 
after  the  substitution  is  made,  and  so  will  he  multiplied  by  r'K 

3.  Def.  A  Rational  Fraction  is  the  quotieiit  of  two 
entire  functions  of  tlie  same  variable. 

A  rational  fraction  is  of  the  form, 

a  -\-  hx  -{-  rx^  -f-  etc. 
m  +  nx  +  px^  -f  etc. 

Any  rational  function  of  a  variable  may  be  expressed  as  a 
rational  fraction.     Compare  §  180. 


■4 


i, 


Equations  of  the  First  Degree  between  Two 

Vjiritibles. 

219.  Since  we  may  assign  to  an  independent  variable  any 
values  we  please,  we  may  suppose  it  to  increase  or  decrease  by 
regular  steps.  The  difference  between  two  values  is  then 
called  an  increment.     That  is, 

Def.  An  Increment  is  a  quantity  added  to  one 
value  of  a  variable  to  obtain  another  value. 


1 

0 


I 


INCllEMEXTS. 


2-25 


to  one 


Y 


IiEM.  If  WO  (liininisli  the  variable,  tlic  increment  is 
negative. 

Tlienrem.  In  a  function  of  the  first  clogroo,  oqiial  in- 
crements of  the  in(l<'])('udent  variable  cauae  equal  incre- 
ments of  the  function. 

Example.  Lot  x  bo  an  independent  variable,  and  call  u 
the  function  -^x  -\-  11,  so  that  wo  have 

If  we  give  x  the  successive  vahios  —2,  —1.  0,  1,  2,  etc., 
and  find  the  corresponding  values  of  the  function  n,  they 
will  bo 

Values  of  r,     —2,      —1,      0,       1,       2,       3,       4,     etc. 


« 


u. 


8, 


it  I,    1),     1;U,  11,     l.V|,   ir,     etc. 

We  see  that,  the  increments  of  x  being  all  unity,  those  of 
y  a  3  all  \\. 

General  Proof.  Let  an  -\-  hx  =  c  be  any  eciualion  of  (lie 
first  degree  between  the  variable  x  and  the  function  ii.  Solving 
this  equation  we  shall  have 

c 


Ix 


^  _^ 

""       a      ~  a      a  ' 


Let  US  assign  to  .r  the  successive  values, 
r,     r  4-  //,    r  -\-  2/i,     etc., 

the  increment  being  //  in  each  case.     The  corresponding  values 
of  the  function  u  will  be 


c       h  ^        c       h  ^       h ,        c       h  ^ 
a      a  '      a      a        "   ' 


9J) 


//,     etc.. 


of  which  each  is  less  than  the  preceding  by  the  same  amount, 

]i.     Hence  the  increment  of  u  is  always h,  which  proves 

a  -^         a  ^ 

the  theorem. 


2*iO.   Gtometric  Construction  of  a  Relation  of  tJie  First 

gree.     The  relation  between  a  variable  x  and  a  function  ii 

of  this  variable  may  be  shown  to  the  eye  in  the  folio  win 


Deg) 


eye 


'ay: 


15 


J'4  ^i 


220 


QKOMETliW  COxMU'Ji  i'VTJOJV. 


•I 


K 


f 


V 


X 


N 


V 


% 


•K 


N 


\ 


V 


•a      -I 


1    ^ 


\. 


N 


X 


N» 


Take  a  base  lino  AX,  mai'1<  a  zero  point  npon  it  and  from 
this  zero  point  lay  olT  any  valucH  of./;  wo  please,  'riion  at  each 
])oint  of  the  line  ('(irresjtonding  to  u  value  of  ,r  erect  a  vertical 
line  e(|ual  to  the  cori'espoiulin;^  value  of  n.  If  ii.  is  positive,  the 
value  is  measured  upward;  if  negative,  downward.  The  lino 
drawn  tiirouudi  the  ends  of  these  values  of  y.  will  show,  hy  the 
distance  of  each  of  its  points  from  the  base  line  AX,  the  values 
of  11  corrcspondinjT  to  all  values  of  x. 

Let  us  take,  as  an  example,  the  e(iuatioii 

5u  +  3.r  =  10, 

3 

the  solution  of  which  lmvcs     w  =  2  —  -x. 

^  5 

Computing  the  values  of  n  corresponding  to  values  of  x 

from  —3  to  -f-G,  we  find  : 


X     —       — «),  ~— rV, 

u=  +3^    +3  J, 


1, 


0  3 


0,  +1,     +2,  +3,  +4,  +r.,  +0. 

2,    Ih    h    I  -I  -1,  -il 


Laying  ((ff  tliese  values  in  the  way  just  described,  we  ha/e 
the  above  figure.  AVberevcr  we  choose  to  erect  a  value  of  u, 
it  will  end  in  the  dotted  line. 

We  note  that  by  the  projierty  of  functions  of  the  first  de- 
gree just  proved,  each  value  of  ti  is  less  (shorter)  than  the  pre- 
ceding one  by  the  same  amount ;  in  the  present  case  by  ^  •     It 

is  known  from  geometry  that  in  this  case  the  dotted  line 
through  the  ends  of  fi  will  be  a  straight  line. 

We  call  this  line  through  the  ends  of  y  the  equation  line. 


OF  EQUATIONS   OF   IIIF   FIIiST  DKGRKK.        227 


-X 


iTul  from 
■n  at  each 
II  verticul 
;itivc,  thu 
The  lino 
\v,  by  the 
[he  values 


lues  of  z 


2'il.  When  wc  can  onoo  draw  this  slrai<^ht  lino,  wo  can 
find  the  value  of  y  corresponding  to  every  value  of  x  without 
using  the  ecimition.  We  have  only  to  take  the  point  in  tho 
base  lino  corresponding  to  any  value  of  q-,  and  by  measuring 
the  distance  to  the  line,  we  shall  Imve  the  corresponding  vahio 
of  tt. 

Now  it  is  an  axiom  of  geometry  that  one  straight  line,  and 
only  one,  can  be  diuwn  between  any  two  points.  Therefore, 
to  form  any  relation  of  the  first  degree  we  ]>Ieaso  between  x 
and  )(,  we  may  take  any  two  values  of  ;/•,  assign  to  them  any 
two  values  of  u  wc  please,  ])lot  these  two  pair  of  values  of  v  in 
a  diagram,  draw  the  ecpiation  line  through  them,  and  then 
measure  olf,  by  this  line,  as  many  more  values  of  y  as  wo 
l)leaso. 

ExAMPLK.  Let  it  bo  required  that  for  u- =  +  I  we  shall 
liave  u  =■  -f-1,  and  for  x  =  +•),  u  =  +  li.  What  will  be  the 
values  of?/  corresponding  to  »•  —  — ',],  — ^,  — 1,  0,  etc. 

Drawing  the  base  line  AX  below,  we  lay  off  from  1  the  ver- 
tical line  -f  1  in  length,  and  from  the  point  5  the  vertical  line 
4-  2.  Then  drawing  the  dotted  line  through  the  ends,  wc 
measure  off  different  values  of  u,  as  follows: 

a:  =  —3,  -2,  -1,      0,  +1,  -j-2,     +3,  +4,     +5,  +0,    etc. 
n  =  -1,  -^,      0,  +1,      1,  -f  H,  +2,  +n,  +3,  +U,  etc. 


,   4-0. 
•1,  -U. 


we 


liave 

ll'ie  of  if, 


,^ 


first  de- 
thc  ] 


ire- 


)V  _■ 


It 


Itted  line 


ion  line. 


EXERCISES, 


1.  Plot  the  equation  2w  -f  dx  =  G. 

2.  Plot  a  line  such  that 

for    X  =:  —  G    we  shall  have    u  =  -{-  4, 
for    X  =  ■\-  6        "         "  w  =  —  4, 

and  find  the  values  of?,  for  a:  =  1,  2,  3,  4,  and  5. 


II 


228 


GEOMETRIC   CONSTRUCTION 


V 

i, 


233.  The  algebraic  problem  corresponding  to  the  con- 
struction of  §  220  is  the  following: 

Ilai'lng  given  two  values  of  y  corresponding  to  two 
given  values  of  X,  it  is  required  to  construct  an  equation 
of  the  first  degree  such  tliat  these  two  pairs  of  values 
shall  satisfy  it. 

Example  of  Solnfion.  Let  the  requirement  be  that  of  the 
equation  jjlotted  in  tiie  preceding  example,  namely, 

for    X  =  1     we  must  have    ?/  =  1, 


for    ^  =  5 


a 


a 


U  =  3. 


The  proljlcm  then  is  to  find  such  values  of  a,  b,  and  c,  that 

in  the  equation 

ax  +  '.'  —  c,  (1) 

we  shall  have  u  =  1  for  r  =  1,  and  u  =  3  for  x  =  5.     Sub- 
stituting these  two  pairs  of  values,  we  lind  that  we  must  have 

axl  +  bxl  =  Cy 
axo  +  l/x3  =  c; 

or  a  +  h   -  c, 

ha  -J-  oh  ~-  ■:. 

Ilere  a,  b,  and  c  are  the  unknowb  'juantlties  whose  values 
are  to  bo  found,  and  as  we  have  0:iiy  iwo  equations,  we  cannot 
find  them  all.     Let  us  therefore  find  a  and  b  in  terms  of  c. 

Multiplying  the  first  equation  by  3,  and  subtracting  the 
product  from  the  second,  we  have 

2a  =  —  2c    or    a  =  —  e. 

]\rultiplying  tlic  first  equation  by  5,  and  subtracting  the 
second  from  the  product,  we  have 


2b  —  4c    or     b 


Substituting  these  values  of  a  and  b  in  (1),  we  find  the  re- 
quired equation  to  be 

2cu  —  rx  =  c. 

Wo  may  divide  all  the  terms  of  this  equation  by  c  (§  120, 

Ax.  Ill),  giving 

2u  —  X  =  1, 


1 


tlic  re- 


(§  1^0, 


\ 


i 


OF  EQUATIONS   OF   THE  FIRST  DEGREE.         220 

thus  showing  that  there  is  no  need  of  using  c.     The  sohition 
of  this  equation  gives 

1  +  X 

"  =  --r' 

from  which,  for  x  =  —3,  —3,  —1,  etc.,  we  shall  find  the  same 
values  of  u  which  we  found  from  the  diagram. 

EXERCISES. 

Write  equations  hetwecn  x  and  y  which  shall  be  satisfied 
by  the  following  pairs  of  values  of  x  and  [/. 

I.  For  X  =  2,   ij  =.  1;  and  for  .r  =  o,  y  =  —  1. 


2. 

3- 
4- 

5- 


For  X  =1  — 


y 


—  1  ;  and  for  x  =  +2,  ?/  =  +1. 


For  X  =  —~),   ij  =.  -\-  2  ;  and  for  x  =   -\-6,  y  =  -—2. 
For  X  =  0,  ?/  =  —  7  ;  and  for  x  =  IT),   y  —  0. 
For  X  =  2^),  1/  z=  2  ;  and  for  x  =  30,  ?/  =  3. 


233.  Gcomcfric  Salution  of  Two  £qtfatlo?is  tcith  Two  Un- 
Jcnown  Quantities.  The  solution  of  two  equations  Avith  two 
unknown  quantities  consists  in  finding  that  one  pair  of  values 
which  will  satisfy  both  equations.  If  we  lay  olf  on  the  base 
line  the  required  value  of  .r,  the  two  values  of//  corresponding 
to  this  value  of  ar  in  the  two  equations  must  be  the  same  ;  that 
is,  the  two  C(iit((tioii  lines  inust  cross  cacli  other  at  the 
point  thus  found.     Ilcncc  the  following  geometric  solution: 

I.  Plot  the  two  equations  from  the  same  base  line  and 
zero  point. 

II.  Continue  the  equation  lines,  if  necessary,  until 
they  intersect. 

III.  Tlie  distance  of  the  j)oint  of  intersection  fronh  the 
base  line  is  the  v(due  of  y  U'liieh  sutisfes  both  equations. 

IV.  Tlie  distance  of  the  foot  of  the  y  line  from  tJie 
zero  point  is  the  required,  value  of  jc. 

EX  ERCISES. 

Solve  the  following  etfuations  by  geometric  construction : 

1.  X  —  2?/  =  3,     2x  +  w  =  o. 

2.  2u  4-  7a-  =  4,     2>u  +  r  —  1. 


f  \"^ 


230 


NOTATION  OF  FUNCTIONS. 


!!;■ 


1  *< 

n 


w       < 


234.  Genmctric  Explanation  of  Equitmhnt  and  Inconsist' 
ent  Efjuatinns.  If  wc  have  two  cquivaloDt  equations  (§  200), 
each  vahic  of  .r  will  give  the  same  vahie  of  the  other  quantity 
n  or  ?/.  Hence  tlie  two  lines  representing  the  equation  will 
coincide  and  no  definite  point  of  intersection  can  be  fixed. 

If  the  two  equations 

au  -\-  bx   =  c, 
a'u  +  h'x  =  c', 
are  inconsistent  we  shall  have  (§  l*i;3), 

b       b' 


a 


a 


If  b  be  any  increment  of  .r,  the  increments  of  ?*  in  the  two 

equations    (§210)  will   be and ,•     Therefore   these 

^  a  a 

increments  will  be  ccpial,  and  the  two  equation  lines  will  be 
parallel.     Hence, 

2\)  iiicoJisi stent  cqiintimis  corrc.ywnd  parallel  lines, 
which  hare  no  pniiit  of  intersection. 

If  the  two  equations  are  equivalent  (§  141,  143),  their  lines 
will  coincide. 

Notation  of  Functions. 

225.  In  Algebra  we  use  symbols  to  express  any  numbers 
whatever.  In  the  higher  Algebra,  this  system  is  extended 
thus  : 

TF b  may  use  any  symbol,  having  a  letter  attached  to 
it,  to  express  a  function  of  the  quantity  represented  hy 
that  letter. 

Example.  If  we  have  an  algebraic  expression  containing 
a  quantity  .r,  Avhich  we  consider  as  a  function  of  x,  but  do  not 
wish  to  write  in  full,  we  may  call  it 

F{x),    or    0{r),     or    [.?•],     or    Ax, 

or,  in  fine,  any  expression  we  please  which  shall  contain  the 

Bynibol  X,  and  shall  not  be  mistaken  for  any  other  expression. 

In  the  fust  two  of  the  above  expressions,  the  letter  x  is  enclosed  in 
parentheses,  in  order  that  the  expression  may  not  be  mistaken  for  3'  mul- 
tiplied by  F,  or  (p.  The  parentheses  may  be  omitted  when  the  reader 
knows  that  multiplication  is  not  meant. 


-  I  i.-nrrm»<ag- 


NOTATION  OF  FUNCTIONS. 


231 


The  fundamental  principle  of  the  functional  notation  ia 
this: 

]]licih  (t  synihul  jcith  a  letter  attached  represents  a 
faiiet'ion,  then,  if  we  sjihstitute  aiiij  other  quant  it  j/  for 
the  letter  attacliecl,  the  eombi nation  will  represent  the 
function  found  by  substituting  that  other  quantity. 

Example.  Let  us  consider  the  expression  ax^  +  5  as  a 
function  of  x,  and  let  us  call  it  ^  {x),  so  that 

0  (.?•)  =  ax^  -{-  h. 

Then,  to  form  <p  (i/),  we  write  ?/  in  place  of  x,  obtaining 

<A  il/)  =  f(f  +  ^• 
To  form  0  (.?;  +  //),  avc  write  x-]-ij  in  place  of  x,  obtaining 

(^{x  ^  y)  =  a  {x  +  /y)2  +  h. 
To  form  (p{a),  we  write  a  instead  of  a-,  obtaining 

0  {a)  =  r/3  +  I). 
To  form  0  (nij^),  we  put  aif  in  place  of  x,  obtaining 

0  (ny^)  =  a  {ny'^Y  +  ^  =  C-^V^  +  ^« 
The  equation  0  (2;)  =  0  will  mean 

az^  +  b  =  0. 

EXERCISES. 

Suppose  0  {x)  =  CLX^  —  (ih:,  and  thence  form  the  values  of 

I.         0(//).  2.         0(,l).  3.         0  {/>//). 

4.      0  (.r  +  v).  5.      0  (,r  4-  «)•  6.      0  (,(r  —  «). 

7.     0(.i'  +  rt//).  8.     <p{x  —  ay).  9.     0(.r2). 

Suppose  i^(.r)  =  art-*^,  and  thence  form  the  values  of 

10.     F(y).  II.     F{2y).  12.     F{:\i/). 

13.     /X-'^  +  //).  14.     F{~^-y)-         15'     ^'Ml). 

Suppose  /  (.(•)  =  .^•2,  and  thence  form  the  values  of 

16.    /(I).  17.    /C''^).  t8.    /(•'•')• 

19.     /(.r").  20.     /(:r=).  21.     f  {x^). 


(* 


11 
snTTi 


232 


FUNCTIONS  OF  SEVERAL    VARIABLES. 


n 


V  ! 


2  2.  Prove  that  if  we  put  0  (.r)  =  a^,  wc  shall  have 

0  (a:  +  ?/)  =  0  (.r)  x  0  (.y),  0  (^7/)  =  [</>  (^^Ol^  =  [0  (Z/)]^' 

Let  lis  put  0  (;;?)  =  m  {m  —  1)  (??i  —  2)  {m  —  3) ;  thence 
form  the  vahics  of 

23.     0((]).  24.     0  (,-.).  25.     0(4). 

26.     0(3).  27.     0(2).  28.     0(1). 

29.     0(0).  30-     '^(-l)-  31-     0(-2). 

Fuiictions  of  Several  Variables. 

220.  An  algebraic  expression  containing  several 
quantities  may  be  represented  by  any  symbol  having 
the  letters  which  rejiresent  the  quantities  attached. 

Examples.    We  may  put 

0  (.r,  y)  z=  ax  —  hj, 
the   comma  hcing   inserted   between   x  and  y,  so  that  their 
product  shall  not  be  understood.     We  shall  then  have, 

0  (/;?,  0})  =  (tni  —  hn. 
0  {y,  x)  =  ay  —  bx, 

the  letters  being  simply  interchanged. 

0  {^  +  U,  X  —  y)  =  a  {x  +  //)  —b{x  —  y) 
-  [a  -  h)  X  4-  («  +  h)  y. 
0  {a,  h)  =  «2  —  52. 
0  (5,  n?)  =  cb  —  ha  —  0. 
0  (r*  +  />,  ^^)  =  n  {a  +  i)  —  ^/i^. 

0  (^,  ft)    =  ft3  —  ^f(^ 

etc.  etc. 

If  we  put  0  (ft,  /;,  r)  =  2a  -\-  3Z*  —  Hr,  we  shall  have 

</>  (;^  ^>  y)  =  -•^'  +  32  —  5^. 

0  {z,  y,  x)  =  2z  +  '3y  —  ox. 

0  {ui.  7)1,  ?».)  =  2?w  +  oin  +  5/»  =  IQm, 

0(3,8,  G)  =  2-3  +  38  —  5-G  =  0. 


Let  us  put 


EXERCISES. 

0  (x,  y)  ==  dx  —  4//, 

f{^'>  y)  =  ^^  +  ^nh 
f{x,  y,  z)  =  ax  +  Z>?/  - 


ahz. 


i 


USE   OF   INDICES. 


2313 


hence 


^). 


veral 


IVlllg 


Tlienco  form  tlic  expressions 


I.     (p{y,x).             2.     (I>{(t,h). 

3- 

•i  (3,  4). 

4.     0(4,3).              5.     0(10,1). 

6. 

f(a,  b). 

7.    f{l>,  a).                8.     f{y,  .r). 

9- 

/(r,  -3) 

lo-  /(y.«  —p)'      II-  /(^'  •*•.//)• 

12. 

f{b,  a,  2) 

13.    f{a,  b,  r).          14.    /(^^^  l/^.  >.^). 

15.    /{—((,   —b,   —ab). 

T   ,           ,      /        s        m  (»i  —  1)  (//<  - 

-•^) 

^        ^       '^       ><■  {ii  —  1)  (/i  — 

•^)' 

Find  the  vahies  of 

16.     (3,3).                 17.     (1,3). 

iS. 

(0,  3). 

19.     ('i,  3).                20.     (r,  3). 

21. 

(8,  3). 

22.     {2,  -1).            23.     (3,  --^i). 

24. 

(4,  -:>). 

their 


Use  of  Indices. 

2'16a.  Any  iiuinber  of  clifT«nviir  quantities  may  be 
represented  by  a  common  symbol,  tht' distinction  being 
made  by  attaching  numbers  or  accents  to  the  symbol. 

EXAMPLES. 

1.  Any  n  different  ({nantities  nuiy  be  represented  by  the 
symbols,  Pi,  2^2^  Pi>  -  -  -  •  Vn- 

2.  A  prodnccr  desires  to  have  an  algebraic  symbol  for  tbe 
amonnt  of  money  which  he  earns  on  each  day  of  the  year.  If 
he  calls  q  what  he  earns  in  a  day  he  may  put : 

<7i     for  the  amonnt  earned  on  January    1, 

.-,  a  a  a  . .  >) 

etc.       '•'  "  "  ''     etc., 

73  1  '^^y 

q^o       "  "  "      February  1; 

and  so  on  to  the  end  of  the  year,  when 

(73  0  5  will  be  the  amount  for  Decemljer  31. 

Def.  The  distinguishing  numbers  1.  2.  3,  etc.,  are 
here  called  Indices. 

A  symbol  with  an  index  attach»'(l  may  represent  a 
function  of  the  index,  as  in  the  functional  notation. 


USE  OF  INDICES. 


bcrs 


EXERCISES. 

Lot  lis  put  at  =  f  {I  -\-  1).     Tlicn  find  the  value  of 

2.  Prove  the  following  e<iUiitioiis  by  computing  Loth  meni- 


a^  +  «3  =  3  f<-i' 


5 


a  I  4-  ((2  +^'3=3  ^3- 

G 

ai  +  a.  +  a^  -{-  a^  =  3^4- 

If  we  put  *Si  =  1  +  2  +  3  .  .  .  .  +  /,  Ave  shall  liave 

S\  =^  1. 

^S'g  =  1  +  3  =  3. 

A'g  =  1  +  2  +  3  —  0,  etc.,  etc. 

Using  the  preceding  notation,  find  the  values  of  the  ex- 
pression.^ : 

5.     2,s'5  —  rtg.  6.     2,S'6  —  a^. 

2211.  Sometimes  tlie  relations  between  quantities  distin- 
guished by  indices  are  represented  by  equations  of  the  first 
degree.     The  following  are  exami)les: 

Let  us  have  a  series  of  quantities, 

vj  Q,     -^11    -^'2'    -^3'    -''^i'    etc.; 
connected  by  the  general  relation, 

Ai,i  =  Ai  +  ylj_i.  (a) 

It  is  required  to  express  them  in  terms  of  A^  and  A^. 

,       We  put,  in   succession,  1  =  1,  i  =  2,  i  =  3,  etc.     Tlien, 
when  1  =  1,  we  have  from  (a), 

A^  =  Ai  +  A„. 

When    I  =  2,        A.^  =  A^  +  A^  =  2J,  +    Jq- 

i  =  3,        A^  =  A^  +  .lo  =  3.1,  +  2Jo. 

i  =  4,         vis  =  yl^  4-  A.,  7=  5.1,  4-  3Jo. 

i  =r  5,         ylfi  =  vlg  4-  ^'li  =  H^li  4-  5 Jo, 

and  so  on  ind(.4initely. 


ir 


MISCELLANEO  US  FUNCTIONS. 


235 


li  mem- 


»f  the  cx- 
es  distin- 


(.) 

A,. 
Ic.    Then, 


0* 
0* 
0' 
0' 


EXERCISES. 

1.  If  Ai.-i   =   Ai  —  Ji^i, 

what  will  he  the  values  of  A„  .  .  .  .  Am,  and  in  what  wa}'  may 
all  siil)sequeiit  values  be  deteniiiued? 

2.  If  Ai,i  =  2Ai-A„ 
find  A 2  to  J 5  in  terms  of  J,,  and  .li. 

3-  If  Ail  =  lAi  4-  Ai_-[,    find  A^  to  A^. 

4-  If  Ai  =  Ai^i  +  h, 

find  the  sum  .f^  +  A^  +  A„  +  .  .  .  .  +  A,t,  in  terms  of  A q, 
h  and  7i.     (Comp.  §  209,  Proh.  V.) 

5-  If  Ai  I  =  rAi, 

find  Jj  -f  Jo  +  J3  +  .  .  .  .  +  ^I,,,  in  terms  of  A^^  and  r. 

6.   If  Ji^  1  =  ii'Ai  +  yli  1, 

find  A»,  A^ Aq,  in  terms  of  Aq  and  Jj. 

Miscellaneous  Fuiictions  of  Numbers. 

238.  We  present,  as  interesting  exercises,  certain  elemen- 
tary forms  of  algebraic  notation  much  used  in  Mathematics, 
and  which  will  be  employed  in  the  present  work. 

1.  When  we  have  a  series  of  symbols  the  number 
of  which  is  either  indeterminate  or  too  great  to  be  all 
written  out,  we  may  write  only  the  first  two  or  three 
and  the  last,  the  omitted  ones  being  represented  by  a 
row  of  dots. 

Examples.  a,  b,  c,  .  .  .  .  t, 

Xf    lif    O)     ....    ^Of 

12  w 

■Am        /W^        •        ■        k       •       tt/y 

n  being  in  the  last  case  any  number  greater  than  2. 
The  number  of  omitted  symbols  is  entirely  arbitrary. 

EXERCISES. 

How  many  omitted  expressions  are  represented  by  the  dots 
in  the  following  series: 


Hv 


11:11 


I  •  J.*      /4>  y      Oy     •      •      •      •      71* 


2*  Xy'^yijytttttl     —     /C* 


2;3G 


MISUELLANEO  US  FUNCTIONS. 


\    i 


n,  li  —  1,  n  —  ^,  .  .  .  .  n 


s. 


n,  it  —  1,  n  —  '^,  .  .  .  .  n  —  .s  —  1. 


3- 
4- 

5- 

6.  )i,  n  —  1,  n  —  '>l,  .  .  .  .  n  —  s  -\-  \. 

What  will  be  the  last  term  ill  the  series: 

7.  2,  3,  4,  etc.,  to  n  terms. 

8.  11,  n  —  1,  n  —  2,  etc.,  to  s  terms. 

9.  2,  4,  G,  etc.,  to  k  terms. 

2.  Product  of  the  First  n  Numbers.    The  symbol 

n\ 
is  used  to  express  the  product  of  the  first  n  numbers, 

1-2-3.  .  .  .n. 

Thus,  1 !  =  1. 

2!  =  1.2  =  2. 

3!  =  1-2-3  =  6. 

4!  =  1.2.3.4  =  24. 
etc.  etc. 

It  will  be  seen  that   2!  =  2-1! 

3!  =  3-2! 
And,  in  general,        n\  =:  n  {n  —  \)\ 

whatever  number  /*  may  represent. 


EXERCISES, 

Compute  the  values  of 
I.     5!  2.     6 ! 


3!  4! 


_8|_ 
^'     3!  5! 


8! 


2w  =  2««! 


6.  Prove  the  equation  2-4. 0-8  .  . 

7.  Prove  that,  when  ;ms  even, 

w,  _  n  {n  —  2)  (w  —  4) ....  4.2 
2  •  ~  ~"         '        M  ■""""  * 

3.  Binomial  Coejjicients.    The  binomial  coefficient 
niji  —  V){n  —  2) ....  to  5  terms 

J../&.o....i9 

is  expressed  in  the  abbreviated  form, 


ibers, 


: 


Ifficieiit 


MISCELLANEOUS  FUNCTIONS. 


237 


c: 


the  parcntlicses  being  usod  to  show  that  what  is  meant 

n 

is  not  the  fraction     • 

s 


EXAMPLES. 

Vol  ~  l'2'3"k'5  ~  " 

CO 
C) 


n 
=  V  =  n. 


_  v{n—  1) 2-1 


m  = 


i-^-a n      ~^' 

(n  +  4)  {>i  +  3)  (;/,  +  2) 
1.2-3 


EXERCISES. 

Compute  the  vahies  of  the  expressions : 


Prove  the  formuke 
5! 


^'     \2/  ~  2!  3! 

5-     U  +  i/  -  s  +  lVs/' 

^  (':)+(;;)=m 


//A    _  71 1 

\s)  ~  s\  {)i  —  s)l 


■  4 


■I'.H 


,11 


B  O  O  K    I  X . 
riFE      TirEORY     OF    NUMBERS 


I     t 
'I 


r  *  iff 
III 


CHAPTER    I. 

THE     DIVISIBILITY     OF     NUMBERS. 

•i'i*.).  Def.  Tlie  Theory  of  Numbers  is  a  branch 
of  niatlieniatics  which  treats  of  tlie  properties  of  integers. 

Def.  An  Integer  is  any  whole  number,  i)ositive  or 
negative. 

In  tlic  theory  of  iuimbcr.s  the  word  mimhcr  is  used  to  ex- 
press an  integer. 

Def.  A  Prime  Number  is  one  which  has  no  divi- 
sor  except  itself  and  unity. 

The  series  of  ])rimc  numbers  arc 

2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  etc. 

Def.  A  Composite  Number  is  one  which  may  be 
expressed  as  a  jn-oduct  of  two  or  more  factors,  all 
greater  than  unity. 

Rem.  Every  number  greater  than  1  must  be  either  prime 
or  composite. 

Def.  Two  numbers  are  prime  to  each  other  when 
tliey  have  no  common  divisor  greater  than  unity. 

Example.  The  numl)ers  24  and  35  are  prime  to  each 
other,  though  neither  of  them  is  a  prime  number. 

Rem.  a  vulgar  fraction  is  reduced  to  its  lowest  terms  when 
numerator  and  denominator  are  prime  to  each  other. 


if 
w 


DlVmniLITT  OF  Nir.V/ih'liS. 


2:jo 


RS. 


branch 

ntegers. 

sitive  or 


iod  to  cx- 
10  divi- 


tors,  all 


icr  pnme 


when 


y 


to  each 
L-ms  when 


i 


Division  into  l*riino  Factors. 

230,  Every  composite  nunibor  may  by  (.Icl'mition  be  di- 
vided into  two  or  inoiv  lUctors.  If  iiiiy  of  tiiese  factors  are 
composite,  tlicv  mav  be  airuin  divided  into  other  factors. 
When  none  of  the  factors  can  be  further  divided,  tlicy  will  all 
be  ])rimc.    Hence, 

Tiii:oiii;m.  Krcry  composite  niunhcr  Diaij  be  divided 
into  fn'iinc  factors. 


KXAMI'LE. 


"Whence, 


180  rz:  i).'Z{), 

9  =  ;m}, 

20  =  4-5  =  2.2-5. 

180  =  2.2-;j.3.5  =  22.32.5. 


Cor.  1.  Because  every  numl)cr  not  prime  is  compositr, 
and  because  every  coin|)(>site  number  may  be  divided  into 
prime  factors,  we  conclude:  Every  uuniher  is  either  prime 
or  divisible  by  a  pj'ime. 

Cor.  2.  Every  nunil)er,  prime  or  composite,  may  be  ex- 
pressed in  the  form 

2)°-q^ry  etc.,  {a) 

where  2^,  q,  r,  etc.,  are  difTerent  i)rime  numbers; 

«,  /3,  y,  etc.,  the  exponents,  arc  positive  integers. 

Rem.  If  the  num1)er  is  prime  chere  will  be  but  one  factor, 
namely,  the  number  itself,  and  the  exponent  will  be  unity. 

EXERCISES. 

Divide  the  following  numbers  or  products  into  their  prime 
factors,  if  any,  and  thus  express  the  nund)ers  in  the  form  {a)  : 

I.     24.         2.     72.         3.     200.         4.     ir.O.         5.     225. 
6.     25G.       7.     91.         8.     143.         9.     300.        10.     217. 
II.     30:2.  12.     1.2.3.4.5.0.7.8.9. 

IiEM.  In  seeking  for  the  prime  factors  of  a  number,  it  is 
never  necessary  to  try  divisors  greater  than  its  square  root,  for 
if  a  number  is  divisible  into  two  factors,  one  of  these  factors 
will  necessarily  not  exceed  such  root. 


ui 


4ii 


240 


vnis/nffjrr  of  numbeu^. 


I 


'I 


Coninioii  Divisors  of  Two  XiiiiiIkts. 

*Z\\\,  'I'hkoukm  I.  //  hvo  funnhrrs  have  a  common 
factor,  tln'w  ftain  will  Ikwc  tliat  same  factor. 

Proof.     Ll'I     a  lie  the  cojiiinon  luctor  ; 

///,  tlio  product  of  all  (he  otlici  lUctons  in  the 

OIK*  number; 
Hi  tljL'  oorrc'spoiuUng  product  in   tlie  other 
innu'ier. 

Then  the  two  numbers  will  l)o 

am     and     an. 

Their  sum  will  be  a  {in  +  //). 

liecau.so  m  and  /i  are  whole  numliers,  />/-f-w  will  also  l)c  a 
whole  numl)cr.     Therefore  a  will  be  a  faclor  of  am -{-an. 

TilFOUKM  II.  If  tiro  iinnilx'i's  hare  a  common  factor, 
their  (lijferencc  will  have  the  same  ftctor. 

Proof.    Almost  the  same  as  in  the  lust  Iheorem. 

Cor.  If  a  number  is  divi.<ible  by  a  factor,  all  niulti[)les  will 
be  divisible  by  that  factor. 

Hem.    The  i)receding  theorems  may  be  expressed  as  follows : 

//  tiro  imnitjers  are  (tivisil)Ie  hi/  the  same  divisor, 
their  sioDi,  it  inference,  and  multiples  are  all  divisiljle  bij 
that  divisor. 

Rem.  If  one  number  is  not  exactly  divisible  by  another,  a 
remainder  less  than  the  divisor  will  be  left  over.     If  we  put 

/>,  tne  dividend; 
d,  the  divisor; 


we  shall  have, 


or 


q,  the  quotient; 
r,  the  rcnuiinder 

/)  —  d(j  +  r, 
J)  —  dq  =  r. 


Example.     7  goes  into  00  0  times  and  3  over.     Hence 


th 


IS  means 


00  =  r-O  +  ,3,     or     00  —  7-0  —  3. 


M. 


1 


DIVISIIUIJTY   OF  NUM/lh'liS. 


211 


rs  in  the 
he  otlu'V 


1  also  l)C  a 

-  an. 

on  fact  or, 


iltiplcs  will 

u.<  TuUows : 

r   (lir'isor, 
risihlc  hU 


iiiiother,  Ji 
'  we  put 


IvLi*.     Hence 


*ilVi,  Pnom.KM.  To  pud  /hr.  ^rrnfrsl  cninwnn  tJirhor 
of  hiut  nn  nihfi'ti. 

Lut  III  and  n  bu  the  mmibcrs,  and  let  ///  be  llie  <,'reaiter. 

1.  Divide  in  by  n.  If  tlie  reniainder  is  zero,  n  will  no  t!io 
divirior  r('(|uired,  becai.sc  every  nund)er  dividcri  itscH'.  If  Ihero 
IS  a  remainder,  let  q  be  the  (|Uotieiit  and  ;•  the  remainder. 

Then  m  —  nq  =  r. 

Tiet  d  bo  the  common  divisor  ref|uired. 
lU'caiise  m  and   ;/   are  bolh   divi.sibjc  by  ^/,  m  —  nq  must 
also  be  divisible  l)y  d  (Theorem  II).     Therefore, 

r  is  divisible  by  d. 

Ueiice  every  common  divisor  of  m  and  n  is  also  a  common 
divisor  of  yi  and  y.     Con\ersely,  ])ecauso 

m  =  ?iq  -\-  r, 

every  common  divisor  of  n  and  ;•  is  also  a  divisor  of  ?n.  There- 
fore, the  greatest  common  divisor  of  m  and  n  is  the  same  as 
the  greatest  common  divisor  of  n  and  ;•,  and  we  proceed  with 
thet;^  last  two  numbers  as  we  did  with  vi  and  n. 

2.  Let  r  go  into  n  q'  times  with  the  remainder  /. 
Then  n  =  rn'  +  r', 

or  n  —  rq'  =  r'. 

Then  it  can  be  shown  as  before  that  d  is  a  divisor  of  r',  and 
therefore  the  greatest  common  divisor  of  r  and  r', 

3.  Dividing  r  by  r\  and  continuing  the  process,  one  of  two 
results  must  follow.     Either, 

u  We  at  length  reach  a  remainder  1,  in  whicli  case  the 
two  numbers  are  prime  ;  or, 

ft.  We  have  a  remainder  which  exactly  divides  the  pre- 
ceding divisor,  in  which  case  this  remainder  is  the  divisor 
required. 

To  clearly  exhibit  the  ])rocoss,  we  express  the  numbers  m, 
n,  and  the  successive  remainders  in  the  following  form  : 
IG 


« 


242 


GREATEST   COMMON   DIVISOR. 


'  f; 


I  (■ 


m  =  n-q  -\-  r, 
n  =  r-q'  +  r', 

r  =  r'-f/  +  r", 

,,,        ,,, 

7     -f  >    J 
etc. 


(r  <  7i) ; 
(r'  <  r) ; 

(/•■'</•); 

r   =;■  .7     +y-,  {r'"  <r"); 

etc.  etc.  etc., 

until  wc  roach  a  remainder  erjual  to  1  or  0,  when  the  series 
term  i  nates. 

EXERCISES. 

I.  Find  the  G.  C.  D.*  of  :?40  and  155. 

Divirlend.       Div.  Quo.    Rom. 
^40  =   155-1  +  85. 

155  =  85-1  +  TO. 

85  =  70.1  +  15. 

70=  15.4+10. 

15  =  10-1  +    5. 


10 


5.2. 


Tlierefore  5  is  the  greatest  co'^mon  divisor. 

Note.     Let  tlio  stiulont  arranjrt"  all  the  followinrr  oxorcisos  in  the 
above  t'orni,  lir.si  dividiiii^  in  the  usual  way,  if  he  liucls  it  uecessary. 

Find  the  La'eatest  common  divisor  of 


2.     399  and  427. 
4.     8  and  13. 


6.     799  and  l'.\32. 
8.     '250  and  G25. 


3.  91  and  131. 

5.  1000  and  212. 

7.  800  and  1729. 

9.  1000  and  370. 

10.  If  ;;  he  a  nnmher  less  than  71  and  prime  to  7i,  show  that 
91  — 2'  i^  '^l^*'  prime  to  7i. 

11.  If  p  1)0  any  nnmher  less  than  n,  the  greatest  common 
divisor  between  71  and  /;  is  the  same  ti5  that  between  71  and 
n  —p. 

12.  If  n  is  any  odd  nnmher,    -  ;^ —  and  —  ,  —  are  l)oth 

2  2 

prime  to  it. 

CoroUaries.  1.  When  two  nnmhers  are  divided  by  their 
greatest  common  divisor,  their  ((notients  will  be  prime  to  each 
other. 


1 


*  The  letterH  (J.  C.  D.  are  an  abbreviution  for  (Jreatcst  Conunon  Divisor. 


tlic  semes 


orciaos  in  the 
I'ssury. 


(>. 


show  tliiit 


p?,s 


lost  COIDIIK^I^ 

wccn  »  aiul 
1  lire  l)ulh 


^m1   liy  tlioir 
•inic  to  eiicli 

iunt)n  Divisor. 


i 


GEARING    OF    WHEELS. 


243 


2.  Conversely,  if  two  numbers,  h  and  //,  j)rinie  to  each 
otlur.  are  cacli  multiplied  by  any  number  (/,  tiien  d  will  be  the 
G.  CD.  of  (In  and  dn'. 

*i.'>.*».  flcariiifj  of  ]\']i((h.  An  interesting  problem  con- 
nected with  the  greak'.st  com- 
mon divisor  is  aU'ordcd  bv  a 
Common  juiir  of  gear  wheels. 
Let  there  be  two  wheels,  the 
one  having  m  teeth  and  the 
other  n  teeth,  gearing  into  each 
other.  If  we  start  the  wheels 
witli  a  certain  tnotii  of  the  one 
against  a  certain  tooth  of  the 
other,  then  we  have  the  questions: 

(1.)  Uow  many  revolutions  must  each  wheel  make  before 
the  same  teeth  will  again  conu'  together  ? 

{'I.)  AVith  how  Mumy  teeth  of  the  on-'  will  each  tooth  of  the 
other  have  geared  ? 

Let  ((  be  {\\v  rei|uired  number  of  (urns  of  the  iirst  wheel, 
having  ///  teeth. 

Lety;  be  the  required  number  (»f  turns  of  the  second,  bav- 
in 2:  n  teeth. 

Then,  because  the  Iirst  wheel  has  m  teeth,  fpn  teeth  will 
have  geared  into  the  other  wheel  during  the  y  turns.  In  ih'.' 
same  way,  pn  teeth  of  the  second  wheel  will  have  geared  into 
the  first.  But  tliesc  numbers  must  be  e(iual.  Therefore, 
Avhen  the  two  teeth  again  meet, 

pn  r=  qm. 

Conversely,  for  every  pair  of  numbers  of  revolutions  p  and 
f/,  wliieii  fullil  tlie  conditions, 

pn  =  qm, 

the  same  teeth  will  come  together,  because  each  wheel  will 

have  made  an  entire  number  of  revolutions.     This  ecjuation 

givi's 

p  _   m 

q  ~  n 


m 


m 


244 


OEAIima    OF    WIII'JELH. 


1)1 


\  «l 


nonce,  if  we  reduce  Ihe  fraction     -  to  its  lowest  terms,  wo 

sliall  liave  the  smallest  number  of  revolutions  of  the  respective 
wheels  which  will  bring  the  teeth  together  again. 

To  answer  the  second  question  : 

After  the  lirst  wheel  has  nuide  q  revolutions,  qm  of  its  teetli 
have  passed  a  fixed  ])oiiit.  Any  one  tooth  of  tlie  other  wheel 
gears  into  every  u^'^  passing  tooth  of  the  Hrsi  wheel.     'JMierefoie 

any  such  toolh  has  geared  into  teeth  of  the  first  wheel, 

that  is,  into;;  teeth,  because,  from  the  last  equation, 

qm 

J—  =  p. 
n 

If  (/  be  the  G.  C.  D.  of  ni  and  n,  then 

1)1  =.  dp, 
n  =  (Iq  ; 


or 


Q  = 


ir 

n 
d 


Therefore  each  tooth  of  the  one  wheel  has  geared  into  only 
every  d*'*-  tooth  of  the  other. 

In  the  figure  on  the  preceding  page,  ?;  =  21  and  n  =  0. 
Hence,  d-=.  3,  and  each  tooLli  of  the  one  will  gear  into  every 
third  tooth  of  the  other.  The  nuni])ers  on  the  large  wheel 
show  the  order  in  which  the  <rearing  occurs. 

llow  long  soever  the  wheels  run,  the  same  contacts  will 
be  repeated  in  regular  order.  Hence,  //  each  tooth  of  the 
one  wheel  must  gear  irith  everij  tooth  of  the  other,  tJiO 
liiLinbers  m  and  n  Diust  be  prune  to  each  other. 

EXERCISES. 

I.  If  one  wheel  has  40  teeth  and  the  other  10,  show  how 
they  will  run  together. 

Show  the  same  thing  for  the  following  cases: 


m  —  72,  n  =  15. 
}/i  —  30,  n  =  25. 


VI  =  24,  n  =  IS. 
vi  =  24,  71  —  7. 


h 


rms,  wo 
>pc'clive 


its  teotli 
(T  whiH'l 
Mioivfoio 

st  wheel, 


into  only 

Ind  »  =  ''• 
into  every 
■  uge  wheel 

h tacts  will 
>li  of  the 
\()lhcr,  the 


show  how 


IS. 

7. 


NVMBEItS   AND    THEIR   DIGITS. 


245 


Relations  of  Numbers  to  tlieir  Dij»its. 

234,  III  our  ordinary  mctliod  of  expressing  nunihers,  the 
second  digit  toward  the  right  expresses  lU's,  the  third  lUO's, 
etc.  IMiat  is,  each  digit  expresses  a  power  of  10  correspond- 
ing to  its  position. 

Def.  The  number  10  is  the  Base  of  our  scale  of 
nuiiienition. 

Note.  The  base  10  is  entirely  arbitrary,  and  is  supposed 
to  have  originated  from  the  number  of  the  thumbs  and  lingers, 
these  being  used  by  ])rimitivc  peoi)Ic  in  counting. 

Any  other  number  might  e([ually  well  have  been  chosen  as 
a  base,  but  in  any  case  wo  should  need  a  number  of  separate 
characters  (digits)  ecpuil  to  the  base,  and  no  more. 

Had  8  been  the  base,  we  should  have  needed  only  (he 
digits  0,  1,  2,  etc.,  to  7,  and  dilTerent  combinations  of  the 
digits  would  have  rei)resenled  numbers  as  follows: 

1  =  1, 

7  =  7, 
10  =  1-8  +  0  =  eight. 
17  =  1-8  +  7  =  lifteen. 
20  =  ::i-8  +  0  =  sixteen. 
50  =  5-8  +  0  =  forty-six. 
234  =  2-82  4-  ;J-8  +  4  =  one  hundred  fifty-six, 

etc. 
Let  us  take  the  arbitrary  num])er  z  as  the  base  of  the  scale. 
As  in  our  scale  of  lO's  we  have 

234  =  2.102  +  3.10  _|_  4, 

go  in  the  scale  of  ^'s  the  digits  2:54  would  mean 

2^2  +  3,v  +  4. 
In  general,  the  combination  of  digits  ahcd  Avould  mean 


o 


a::^  +  bz-  +  cz  +  d. 


I)i\  isihilily  of  Nmiibers  aii<l  ihv'w  Dibits. 

*^I55.  'ri[H()iii:Nr.  Tf  the,  sum  of  the  (li<:>ifs  of  niiii  luiin- 
hcr  he  sul)lr(iclc<l  j'roin  the  nuiuber  itself,  the  rciiuiimlcr 
will  he  dluislbic  bij  z  —  l. 


240 


DIVISIBILITY   OF  NUMBERS. 


n', 


)\\ 


Proof.    Let  the  digits  be  (u  h,  c,  iL    Tiie  number  expressed 

uz^  +  hz^  +  cz  -\-  (I 
n     +  Zi    -{-  c   -\-  d 


will  be 

Sum  of  diixits 


Subtracting,     rem.  =  a{z^—\)  +  b{z^—\)  -\-  c{z—i). 

The  factors  z^'  —  I,  r  —  1,  and  z  —  \  arc  all  divisil)le  by 
2;  —  1  (§  U;]).     Hence  tlie  theorem  is  proved.     (§  ;il31.') 

Tii  1:011  KM.  Ill  (iiiy  HCdlo.  havui<2  z  an  its  hast,  tlie  SJini 
of  the  (limits  of  (Uiij  nitnibcr,  irJtcii  dirided  by  z  —  1,  will 
Icava  the  same  remiilndcr  as  irill  tlie  niuubcr  itself  icheib 
so  divided. 

If  we  put:      n,  the  number;     s,  the  sum  of  the  digits  ; 
r,  r',  the  remainders  from  dividing  by  z  —  1; 
fj,  f/,  the  quotients  ;    we  shall  have, 
Number,  n  =  q  {z  —  I)  -{-  r 

Sum  of  digits,    s  =  <f  {z  —  1)  +  r' 
Ilemaindcr,  (7  —  q')  {z  —  1)  -\-  r  —  r . 

Because  u  —  s  and  {q  —  q)  {z  —  1)  are  both  divisible  l)y 
z  —  1,  their  diiference  r  —  r'  must  be  so  divisible.  Since  r 
and  r'  arc  both  less  liian  z  —  1,  this  remainder  can  be  divided 
by  z  —  1  only  when  r  =  r',  which  proves  the  tlieorem. 

Zero  is  considered  divisible  by  all  numbers,  because  a  re- 
mainder 0  is  always  left. 

If  a  bo  any  factor  oi  z  ~  I,  the  same  reasoning  will  api)ly 
to  it,  and  therefore  the  theorem  will  be  true  of  it. 

In  our  system  of  notation,  where  z  =  10,  the  above  theo- 
rems may  be  i)ut  in  the  following  well-known  form: 

7/  /lie  sniii  of  the  di<Jitsof  any  imiidtcr  Ite  divisible 
by  fj  or  0,  the  iiuuiber  itself  will  he  so  divisible. 

These  are  the  oidy  numbers  of  which  the  theorem  is  true, 
because  3  is  tlie  onlv  divisor  of  U. 

TilKoUKM.  fj'  from  any  nii inbrr  ire  subtrael  tJir  di'Ji/s 
of  the  even  ])owers  of  z.  and  add  f/iose  of  the  alternate 
powers,  tlie  residb  will  be  divisible  by  2+1. 


J'ru'jf.     To 
Add 


az^  +  bz"^  ^-  ez  -f  d 
a     —  I)     -I-  r   —  d 


Kesult,    a{z^-\-\)  +b{z^—i)  +  c{z  +  l). 


pre<?-cd 


iWAc  by 

he  SI  nil 
.  1,  irill 
f  when 


in 


■isible  1)y 
1  SSiuce  r 
(livuk'd 

use  a  rc- 
ill  apply 

ovc  tlico- 
Jirisihlo 
1  is  true, 

Itcnnifo 


\\ 


NUMBERS   AXD    Til H Hi   DIGITS. 


247 


Tlic  factors  of  a,  h,  and  c  arc  all  divi<ililc  by  0+1  (^§  !):», 
94),  whc'iico  tho  result  itscli'  is  S'i  divi-siblo. 

Applying  this  result  to  tbo  case  of  z  =  10,  we  conclude: 

//'  oiv  siihtrdctiiig  the  siini  of  the  dhjits  iti,  the  jihira 
of  units,  hundreds^  tens  of  tlioufntuds.  etc..  from  the  sti iih 
of  tJie  (ilteriKite  oiicx,  the  remaimler  is  dirisible  Itij  11, 
the  nuinher  itself  is  divisible  Ijij  11. 

ir  )n  be  any  factor  of  z,  it  will  divide  all  the  terms  of  Iho 
number 

(cc"  +  hz"-  +  cz  4-  (h 
except  the  last.     Hence,  if  it  divide  this  last  also,  it  ^vill  di- 
viile   the  numl)er  itself.     Applying  this  result  to  the  c;ise  df 
2;  =  10,  we  conclude  : 

If  the  hist  di  1^1 1  of  (in  11  iniDihev  i^i  dirisUtle  Inj  o  fue- 
tor  of  10,  the  niunfjer  itself  is  dirisible  by  that  farttn\ 

The  factors  of  10  beinij  'i  and  5.  tliis  rule  is  true  of  these 
numbers  <»nly. 

It  will  be  remarked  that  if  the  base  of  the  system  had  \)Qv\\ 
an  odd  number,  w^e  could  not  have  distiniruishcd  even  and  odd 
nunu)ers  by  their  last  (igure,  Uo  we  habitually  do. 

For  e.\ami)le,  if  the  base  had  been  9,  the  figures  T"-i  would 
liavc  rei)resented  what  we  call  sixty-five,  which  is  odd,  and  73 
"Would  have  represented  what  we  call  sixty-six,  which  is  even. 

The  use  of  the  base  10  makes  it  easy  todt-tect  when  a  num- 
ber is  divisil)le  by  either  of  the  first  three  prime  numbers.  2,3, 


an 


d  5.     If  the  hist  figure  is  divisilile  bv  'i  or  .5.  the  whole  n 


um- 


H'r 


so  divisible 


To  ascertain  whether  3  is  a  factor,  we  find 


whether  the  sum  of  the  digits  is  divisible  l»y  3. 

Ill  taking:  the  sum,  it  is  not  necossary  to  inchid«'  all  the  dibits,  hut  in 
nddinjr  we  may  omit  all  ;5's  and  i)'s,  and  drop  :»,  0,  or  U  from  the  (•uiu  as 
ol'ti'n  as  coDviMiicnt.     Tims,  if  tlic  iiumhiT  were 

5»21(i4-271-2, 
wo  slioiild  pcrl'onn  the  o|)(M':it'ou  mentally,  thus: 


Drop  J) 


1 


w 


hicli  dri>n  ;    (5.  drop  :    4  -»-  2  —  0,  which  drop; 


il" 


1  ^  8  +  3  —  10,  which  leaves  a  remaindtT  1. 


EXERCISES. 

1.   Prove  th.at  if  ;in  even  number  haves  a  remainder  1  when 
divided  by  3,  its  half  will  leave  u  remaiuder  'i  when  jjo  divided. 


218 


DIVISIBILITY   OF  MWIDEliS. 


2.  If  from  any  number  Ave  subtract  tbe  sum  of  units'  digit 
])lns  the  j)r()(lu{'t  of  tlie  tens'  digit  by  /,  i)Uis  the  product  of 
the  iumdreds'  digit  by  €-,  etc.,  tlie  reniaiinU'r  will  be  divi.siblo 
by  10  —i.     {i  may  be  any  integer,  jjositive  or  negative.) 

NoTK.  When  i  —■  1,  this  gives  tlie  rule  of  O's  and  when  t  =  —  1,  tho 
rule  of  ll's. 

Prime  Factors  of  Numbers. 

^,*5(>.  FiiisT  FuxDAMENTAL  Thkoukm.  ,i  jn'oduct  CCUl- 
nob  bo  divided  hij  (i  prime  iiinuhrv  unless  one  uj'  tJie  fac- 
tors is  divif<ihle  hij  tluit  jtriine  niunhcr. 

Note.  Tliis  theorem  is  not  trne  of  composite  divisors.  For  exam- 
ple, neither  8  nor  S)  is  (UviHil)k!  l)y  (»,  but  the  protUict  8.9  =  72  is  bo 
divisibh?.  But  if  we  take  as  many  lUinilMTS  as  we  jih'asi^  not  divisihh.'  by 
7,  we  shall  ahvay.s  fintl  their  pnxluct  to  leave  a  remainder  when  we  try 
to  divide  it  by  7. 

To  make  the  demonstration  better  understood,  we  shall  first  take  a 
special  case : 

The  product  GGrt  is  not  dlcisibh  hy  7,  unless  a  is  divisible 
Inj  7. 

Proof.     Su])posc   ..........     nOr/ div.  l)y  7 

7  goes  into  00  '.)  times  and 'J  over,  liecausc  7'0  =  G.'J,  O;)^^  div,  by  7 
Therefore,  by  Theorem  II,  §  ;231, "aaTivTby? 


I 


•) 


3  goes  into  7  2  times  and  1  over.     ^lultiply  by  2,     iUi  tliv.  by  7 

Subtracting, 7rt  div.  by  7 

We  have  left, «  div.  by  7 

Hence,  if  OOa  is  divisible  by  7,  then  a  is  divisible  by  7. 

Gauss's  Dcmniistraiinn.  If  it  he  possible,  let  am  be  Ihc 
smallest  multiple  of  m  which  is  divisible  byy>,  when  neither  a 
nor  in  is  so  divisible.  If  a  is  greater  than  j),  then  let  p  go 
into  a  b  times  and  r  over,  so  that 


or 


a 


a 

bp 


=  f'P  +  r, 


=  ;•, 


Then, 
Subtract 
Ilemainderj 
Or 


a)n  m\. 


in 


?v^^_ 

{a  —  bp)  ni 
rni 


by  /). 


it 


« 


PlilM/'J   FACTORS   OF  NUMBFRS. 


240 


^'  digit 
hut  of 
ivi^iblo 

-1,  tho 


ict  can- 
he  fac- 
tor pxani- 
-  72  it*  K<> 
ivisil)h;  l)y 
(Ml  we  try 

irst  take  a 
(  dirisiblc 

(liv.  l)y  1 
mv/by  7 

TinTbyl 
(livjjty  7 
div.  i)y  7 

lu  l>o  tlic 
lucitlu'V  a 
let  /J  go 


That  is,  if  a/«  is  divisible  by  p,  so  is  r/it,  where  /•  is  less 
th;iii  jK 

Tlierefore  the  siii;illest  inultiple  of  m  wliieh  fiillils  (he  con- 
ditions must  be  less  than  pni. 

Therelbre,  let  a  <  p.     Let  a  go  iuto^;  c  times  and  .v  over, 

so  that 

p  z=  ca  -{■  s, 

p  —  ca  =  s. 


or 


Then 


Siibtraeting, 


pm  div.  by  p. 


cam 


Or, 


{p  —  ((()  III.    " 


"     (by  hypothesis). 


Therefore,  .s>  being  le?s  th:in  a.  a  is  not  the  smallest  muKiple; 
Avhenee  the  hy[)othe.sis  that  a  is  the  smallest  is  impossible. 

General  Dciiiun.sfrdiion.     Suppose 

p,  a  prime  numlter  ; 
a,  number  not  divisible  hy  p; 
am,  a  product  divisible  by;;. 

We  have  to  prove  that  in  must  be  divisible  by  p. 

Tjet ;;  go  into  r^  f/  times.     Hecauso  a  is  not  divisil)le  l»y7>, 
a  remainder  r  will  be  left.     That  is. 


a  =  jiq  4-  r,     or    a  —  pq  =  r. 


Let  ;•  go  into />  7'  times  and  leave 


am  div.  by  p. 


a  remaimler  / 


Then, 


p  —  q'd  -f  r', 

and  because  pm  iind  n'rm  are  hoth  di- 
visible by  p,  rm  is  so  divisible. 

\\\   the  same  wiiy,  if  ;•'  goes  into  p 


iiider 


7    times,  and   leave  the  remai 

I'  III  will  be  divisible  by/;.     Siiu'i- each 


pi  I  III 

nil 
fj'rm 

])lh 

r'm 
q  r  III 


« 


« 


(( 


<( 


; 


im. 


ot'  the  remanu 


Irrs 


/•.  /■  .  /• 


el( 


mil 


r  III 


be  less  tiian  the  |)recediii^.  we  shall  at 
length  reach  a  remaimler  1,  which  will  give 

VI  divisible  by  p.        Q.  E.  D. 


250 


DIVIsmf/JTV  OF  NUMDERS. 


'I 


/■y/('//si())/  1(1  Srrcnil  F((c//irs.  If  ;//  is  n  in'odiict,  b  x  a,  anil 
/>  is  M(»t  ili\  isihk'  hv  />,  tlicii  wc  iiinv  show  in  tlii'  ^uinc  wav  that 
n  iiuist  be  .so  divisildi'.  If  ti  =.  rs,  and  r  is  not  divif«iljU',  tlit-n 
A'  must  1)0  divisildo,  and  so  on  to  uny  ninuljcr  oi!  factors. 

lion  CO,        ,        • 

TllliouiiM.  fj'  ff  lii'odiicl  iif  (I iiij  III! uiIk'I' of  J'dchii's  /.v 
(lirisihlf  liij  ii.  iiiiiiic  nitiiihci',  Uicii  one  oj'  the  j\(cl(/rs 
must  hi'  il'n'i>>iblc  by  flic  stintr  prime. 

This  Ihi'oivm  is  the   lo'doai    ( .|iii'alont    of   the   one  just 


cnunoiatod  as  the  lirst  finu 


'A\ 


thoor 


oni. 


XoTK.     'I'lu'  Student  will   rcis 


\\\\^'      u>  ])r('('ciliii,<j  (IcnKiiistriitKtii 


Icni 


ni»i)l 


])])lii's  ((Illy  when  tlif  divisor  /)  is  n  ])riiiu'  i.     .mmm". 


I  fit 


were  coiiiposiU' 


Avc  iiii<;lit  rciuli  11  rciniiiidir  uliicli  wuuld  exactly  divide  it,  and  tluii  tlio 
conclusion  would  not  t'((ll(>w. 

2oT.    Sf.COXI)    FrXDAMKNTAL    TimoKENr.       ,/    iiiimhcv 
C((ii  he  (/irif/ff/  iiiti)  pri inc  J'ticlors  in,  onl ij  one  i<'(i\i. 

Fof,  sii])[»oso  wo  could  express  the  number  ^  \\\  the  two 


ways  (§  'Xy>\,  Cor.  'X), 


N  =  )f-  r/  ;-v, 


N  =■  (f^  h 


't'  ('; 


where 7^  q,  r,  etc.,  r^  h,  r,  etc.,  are  till  prime  numbers.     Then 


j,'^  qli  )••/ 


(f^  Ij'' 


If  common  prime  iaotitrs  iiitpiarod  on  both  sides  of  this 
0([Uiition,  wo  could  divide  them  out,  loa\in^  ;m  o<iuation  in 
which   the   prime  factors  p,    /j,   r,  etc.,  are   all   dillViciit    from 


/; 


il,   II.   ('.   CIC 


Th 


ii'ii,  hccause  II,  0,  r,  c 


di\  i>il>lc  by  />. 


TI 


'tc,  tire  all  prime,  none  (d"  them  are 

ttd  theorem. 


lorefore,  bv  the  lirst  i'uiidamen 


llicir  products  cannot  be  so  divisilde.  Hut  the  left-haml  ntem- 
bcr  (d'  the  ('([uatioii  is  divisible  by  />,  because  ^ms  one  of  its 
factors.     Therefore  the  e<piation  is  imi>ossilde. 

Ki;  ,1.  This  theorem  forms  the  basis  of  the  tlieory  of  the 
divisibility  of  numbers. 

'i'he  precodiiii^  theorems  en:d)le  us  to  place  tlie  definition 
of  iiundjers  prime  to  each  other  in  a  new  shape. 


V.     f 


BISOMI.  1 L    COEFFK 'JhWJS. 


251 


I,  anil 
y  1 1  lilt 


'aclois 

no  ju.4 


ustnition 
thtii  the 


nmnhcr 

II  • 
the  two 


..     Then 


() 


f  thi? 


Iiation    in 
nl   l'i"«»i>i 


them  :nv 


ti 


U'DlVIH, 


liul  luc'in- 
)uc  of  it- 


Two  iiuiiibcis  arc  said  to  1)C  prime  to  each  other 
wliuii  they  liavc  no  coiniiioii  ])rinie  I'actois. 

KxA.Mi'Li:.  If  one  nunil)er  is  p'^ifv'*,  iiiul  the  oflier  is 
(if^  l)'r"  (/>,  (/,  r,  etc.,  and  a,  h,  r,  etc.,  beini;;  prime  minilHTs), 
then,  it'/^  y,  /',  etc.,  arc  all  dillVrcnt  from  (/,  b,  c,  etc.,  the  two 
numbers  will  be  i)rime  to  each  other. 

l']UMii('iilar.v  TliconMiis. 

?ii»S,  The  I'tillowin^j^  fi^eneral  tlieorems  I'ollow  IVom  (he  two 
]>reee(lin_ij^  finnhimental  theorems,  and  their  demoiistrution  i:i 
ill  part  left  as  an  exercise  for  the  student. 

I.  A))  /xiirri'  of  an  irrt'duciblc  viilijar  fi'dction  cim  be 
a  ii'liole  nmiihrr. 

Nun;.  An  irreducible  vulgar  fraction  is  one  Avhich  is  re- 
duced to  its  lowest  terms. 

II.  CouoLLARV.  Xo  root  of  (t  icliolo  niLiiihcr  ca. ^  he  a 
vitl^tir  fractioti. 

III.  //  a  niiinhrr  is  dirisihJc  Jnj  srrcraJ  dirhors,  nil 
prime  to  each  other,  it  is  also  (d risible  by  their  product. 

Cor.  To  prove  that  a  numl)er  X  is  divisible  by  a  number 
B  =1  p"-'!^  ry.  it  is  sutliclent  to  prove  that  it  is  divisible  sepa- 
rately l)y  })",  l»y  (f\  by  /-v,  etc. 

Example.  If  a  number  is  divisibk-  separately  by  5,  8,  and 
D,  it  is  divisible  by  a- 8-1)  =  Ij(j(».  lleiice,  to  jn-ove  that  a  num- 
ber is  divisible  by  o<J0,  it  is  suilicient  to  show  that  5,  8,  and  '.» 
are  all  factors  of  it. 

IV.  If  llir  iiu})}'''(ifi)rtiiid  denoDiinfttor  of  ft  I'ahjar 
fraction  hare  no  c.'iuinon  prime  factors,  it  is  reduced  to 
its  lowest  terms. 


f  tlic 


rv  o 


llelinition 


l>iiioiiiial  Coc'lticioiits. 


'i.*>l).  Tlirorcin.  The  ])r()dii('t  of  any  n  consccntivi? 
numbers  is  divisi])lo  \^y  tlio  product  of  the  numbers 
l-2;3  .  .  .  .  n,  or  //  ! 


liil' 


252 


BINOMIA  L    Vl)EFFlVH':yriS. 


'I 


Rem.  The  tlicorcm  implies  that  all  hinomiiil  eocfricicnts 
arc  whole  nuinhers,  becuusie  they  are  quotients  formed  by  di- 
viding the  proihict  of  y*  conseeiitive  niwnbers  by  n\ 

Proof.     1.   We  have  lirst  to  liud  the  prime  iaetors  of  the 

product 

1-2-3-4.5.0 n  =  n\ 

bc^nniiing  with  the  factor  5i. 

1.  The  numbers  divisible  by  3  are  the  even  numbers  2,  4, 

0,  etc.,  to  H  or  )i  —  1,  tiie  number  of  which  is 


n 
2 


Note.     The  expression 


« 
i 


here  means  the  (jrcalcd  whole 


nvmher  in    ,,  which  is  ^  itself  when  n  is  even,  and  — ..  - 
when  u  is  odd. 

The  qiujtients  of  the  division  arc 

1,    tiy     Of    4;    •    .    •    . 


n 
4 


Of  these  (jiioticnts, 
second  set  of  ({uotients, 

1,    -v,    o,    . 

The  next  set  of  quotients  will  be 


are  divisible  by  3,  leaving   the 


n 
4 


1     2 


n 

8 


The  process  is  to  be  continued  until  we  have  no  even  num- 
bers left. 

Thereloi'c,  if  wo  put  a  f<»r  the  number  of  times  that  the 
factor  3  enters  into  ;/ !  we  liave, 


■  -  CI 


-f 


H 

4 


+ 


// 
8 


+  etc. 


II.  The  numbers  in  the  series  )i !  containing  3  as  a  factor  are 

3,  0,  y,   13,  etc., 


I! 


niNOM/A  L    COEFFICf/JNTS. 


253 


in  num- 

luit  tho 


lictor  arc 


of  wliich  tlio  nnmbor  is  •    TIio  (niotients  ol)taiiicil  by  di- 

vkliiig  (hum  by  3  uro      •-.'  J 


1,  2,  3,  ....P^ 


or  these  (luotieiits 


n 


J 


are  a'jjaiii  divisible  by  '.],  jiiid  so 


oil  as  bc'I'ore.     llenee,  if  we  put  /i  for  the  nuniber  of  times  til 
contains  .'J  as  a  factor,  wc  iiavo 


P  = 


+ 

n 

+ 

n 

'w  1 

-f  etc. 


+ 

11 

+ 

In  tho  same  way,  if  h  be  any  prime  number,  n\  will  con- 
tain k  as  a  factor 


+  etc.  times. 


Note.  This  elegant  process  enables  us  to  find  all  (he  prime 
factors  of  n\  without  actually  computing  it,  and  thus  to  ex- 
hibit n\  as  a  product  of  prime  factors.  If  we  su])pose  n  =■  I'lj 
■we  shall  lind, 

12!  =  1.3.3 13  =  310.35.52.7.11. 

3.  Next  let  us  find  the  prime  factors  of  the  i)roduct 


I 

{a  +  l)(a  +  •>) («  +  n), 


F""! 

rn  "I 

4- 

J'_ 

Ly'J 

which  contains  11  factors.  Dividing  successively  by  3,  3,  5,  7, 
etc.,  it  is  shown  in  the  same  .vay  as  before  that  the  prime  fac- 
tor 7^  is  contained  in  the  product  at  least 


+  etc.  times, 


whiitcver  prime  factor  />  may  be.  'IMierefore  the  numerator 
{<(-{- \)  ('/  +  'v) .  .  .  .  (a  -\-  ii)  contains  all  the  prime  factors  found 
in  ;/ !  to  al  least  the  same  [lower  with  which  they  enter  /i\ 
Hence  (§  338,  III),  the  numerator  is  divisible  by  /t! 

Cor.     If  the  factor  (( -{- n   in    the   numerator   is  a   prime 
number,  that  prime  cannot  be  contained  iu  nl  because  it  is 


m 


n 


254 


niVlSOliS    (iF    .1    JSUMUKli. 


»l 


'I 


|y  III 


prcadT  timn  n,     IUiku  the  binuiniiil  I'uutor  will  be  UivisilWe 

by  it.  • 

r.  <t  "* 

KXA.MI'Li;.       ,      ,    .,    IS  (llVlSlblo  l»v  7. 

\Vc  iiwiy  show  in  the  wuiic  way  tliat  the  biiinniial  ((u'lUciciit, 
is  divisible  liy  till  the  itriine  iiuiiiIkts  in  its  uumcrutur  which 
exfcctl  n. 

Divisors  of  a  Xiunlu'r. 

*iM).  I)<f.     The  cxiuvssioii 

0  im) 

is  used  to  express  how  inany  nuinbcTS  nut  greater  tliiin 
Qii  iuv  i)riiiie  to  m. 

Example.     Let  us  find  tlic  viduc  of  </>(r»). 

1  is  i)rinio  to  t),  boiausc  their  G.  CD.  is  1. 

2  «  «  '<  «  a  (( 

'.\  is  not  prime  to  9,  because  their  G.  C.  D.  is  3. 
4  is  prime  to  0. 

o 

0  is  not,  because  U  and  I)  have  tiie  G.  C  D.  3. 

7  is. 

8  is. 

U  is  not. 

Therefore,  the  numljers  less  than  0  and  prime  to  it  are 

J,    /.,   4,    ij,    <,    o, 

which  are  six  in  number,     llt'iiec, 

Till'  numbers  less  than  l'^  and  j)rime  to  1^  arc  1,  5,  7,  11, 
Hence, 

0(l->)  =  4. 

"We  find  in  this  way, 

0(1)  =  1,  0(:i)  =  1,  0(3)  =3, 

0(t)=2,  0(5)  =4,  0(0)  =.-.>, 

0(7)  =  6,  etc.,                          etc. 


o 


i)ivi.<ons  OP  A  M')fni:R. 


2r)5 


•  Cor.  I.     TIic  iiimiltcr   1    is  |»rinK'  to   itsi-lf,  hiiL  no  oilier 

huiiiIkt  i-i  priiiu'  to  itself. 

Cor.  'I,     If  in  he  11  jiriiiK'  inmilter,  then 

0  [in)  —  in  —  I, 
heeaiisc  the  iiunihcrfi    I,  'i,  '.i,  .  .  .  .  ni  —  I    arc  then  ;ill  priino 

to    ///. 

'I'lie  f'ollowiii;^  reniarkahle  theorem  is  assoeiated  with  the 
fund  ions  <l>  (///). 

*ill.   Theorem.     If*  iV"  he  any  nuiiilx'i',  ;iii(l  ff^.d.^, 
(/■^,  etc.,  all  its  divisors,  unity  and  n  inclndcd,  thuu 

(^j  (//i)  +  <l^ ((/.,)  4-  (}>{(l.^)  +  etc.  —  N. 
Faamim.i:.     Let  the  numhcr  he  18. 


-  ? 


■  "■. 


Till!  lUvijjurs  arc  1,  :i,  o,  <;,  (),  18.     Wu  lind,  by  couiilin'^ 

0(1)     =     1 
0(-2)     =    1 

0((;)    =    i 

0(18)  =0 
Sum,     IS. 

To  show  how  this  comes  altout,  write  down  the  numl)ors 
1  to  IS,  iiiid  uiidtTeaeh  write  the  greatest  common  divisor  of 
thill  numher  anil  IS,     Thus, 

Num..   1  -i  '\  i  :.  <;  :  8  i»  10  II  \-i  v.\  11  i:.  lo  ir  is. 

(;.(J.l).,  1  ;i  ;]  ;i  1  G  1  ;i  D    ^     1    G    1    I    ',\    'i     1  IS. 

Necessarily  lli  mimhers  in  the  second  line  are  all  divisors 
of  18  as  well  as  of  the  nundiers  over  them. 

The  divisor  1    is   under  all  the   lunnhers  })rime  to  18,  so 


that  there  are 


0  (IS)  —  divis<U's  1. 


n 


18 


If  )i  he  anv  numher  over  the  divitjor  )l,  then    '  and     .  ,  <u' 
l>,  must  be  prime  to  each  other.     (§  'iWl,  Cor.  1.)     That  is,  the 


2D0 


LlViaOnS  OF  A   NUMBEU, 


•■I 
Iffl 


)   \ 


iiiinilitTs  ?'  arc  all  those  whicli,  wlicn  ilivuled  by  2,  arc  primo 

to  9.     So  there  arc 

0(t»)  divisors  JJ. 

The  divisor  ?)  marks  all  minibers  which,  Avhcii  divided  by  ;}, 

arc  prime  to  -,.-  =  0.     Hence,  there  arc 

0  (<))  divisors  3. 

In  the  same  way  there  are  0  (•'})  divisors  G,  <p  (•^)  divisors  0, 
and  0(1)  divisor  IS. 

The  total  numl)er  of  these  divisors  is  both  18  and  f/>(lS) 
+  0  (!))  +  etc.     Hence, 

0(lS)  +  r>(9)  +0(r,)+0(;i)  +  0(-.>)  +0(1)   z:.   IS. 

General  Proof.    Let  m  be  the  f,nven  number; 
r/|,  f/o,  (1^,  etc.,  its  divisors; 

'7i>   Vs'   '/a'  t'"-'  'inotk'i.ts  ^^  ,  ^^-,  etc. 

The  f|nntients  i^j,  (^o,  etc.,  v.ill  be  the  same  numbers  as  J,, 
^7o.  etc.,  only  in  reverse  order.  The  smallest  of  each  row  will 
be  I  and  the  great-'st  di.     We  shall  then  have 

m  —  (/i  //,  =  (l.^  72  =  </3  73,     vie. 

From  the  list  of  nujubers  1,  'i,  3,  .  .  .  .  ?^/,  select  all  those 
Avhieh  have  </,  (unity)  as  the  <rre:itest  common  divisor  with  ///, 
tlien  those  whicli  have  fi.j  as  such  common  divisor,  tlien  those 
MJiich  have  ^/j,  etc.,  till  we  reach  the  last  divisor,  which  will 
be  7n  i'self,  and  which  will  correspond  to  ni. 

The  iiiinibcrs  haviiiLT  unity  ;'.;  G.V.  \).  will  be  those  prime 
to  ///,  by  de!iiiilioii.     'I'heir  numb(  r  is  0 (///). 

Those  haviiifr '7-  Ji-^  d-V.  1).  with  )/i  will,  when  di\idi'd  by 

(U'  'A'^'^'  <|Uohiiils  prime  to    .     or  to  '/.,.     .Mori'ovcr,  such  <|no- 

liftits  will  iiicliidi'  all  the  uumbers  h<if  ^'rcaltT  lliaii  </.,  and 
](rimi' lo  il.  because  earh  of  tlM'>e  iiuudK'rs,  wIk'Ii  mulli|ilicd 
by  (f.y,  will  ^nvc  a  numk-r  not  jLTn-ater  I  ban  •/.  and  haviu.i;  (/., 
as  its  CJ.  C.  D.  with  »n.      Hence  the   numitcr  <»f  numbers  not 


FI'JliMA  T'S   TIIhVlih'Af. 


257 


)nnic 


.1  by  '^, 


isors  0, 
18. 


•i)\v  will 


ill  thnso 
with  nl, 
M\  those 
I  it'll  will 

|e  ]irinic 
I  ill  I'd  l)y 

\r\\  '|lii)- 
7.,    il'lil 

Itiplicil 
Iviiiii"  '/.J 
Lr.s  i»<)t 


frr 


vater  than  7;?,  and  havinir  (L   as  its  (J.  CD.   witJi  m  will 


be  (}i  {'/.,). 

Continuiii*:^  the  process,  avc  shall  roach  the  divisor  ///,  which 
will  have  m  itself  as  (J.  CD.,  and  which  will  count  as  the 
n II ni her  correspond in<j;  to  (/>(1)  —  1  in  the  list. 

The  tn  niinihers  ],  'rl,  J), .  .  . .  m  are  therefore  e(jual  in  num- 


ber to 


<PM  +  <P{'/,)  +  '/'(73)+  ••••  +0(1) 


or.  since  the  (|U()tients  and  divisors  arc  the  stune,  only  in  re- 
verse order,  we  shall  have 

<^(1)  +  "X'^i)  +  9{'^i)  + +  0(w)  =  m. 

*iVi.  Fkumat's  TirKoitKM.  If  p  hr  ((iiij  f»rinir  iiiiiti- 
hrr,  find  <i  he  a  luiiiibcr  prlniG  to  jh  then  iiP~^  —  1  u-Hi  bo 
Wiviaible  hij  p. 

Ex.VMi'LKS.     (i^  —  1  is  divisible  bv  5  ;  rt"  —  1  is  divisiblchv  7. 

Proof.     Develop  ni^  in  the  l"(»ll<»wiii<,'  way  hy  the  hinoniial 


tl 


K'oreni, 


a' 


[l  +  (a-l)P 


=  1  +/>(^<-J)  + 


(!')  («-.)= 


-f 


•    •   •    • 


-\-iH-  !>' 


Jieciiuse  p  is  i)rinie,  all  the  binomial  cocllicientr 


etc.,     to 


t^-.)' 


ar 


V  (livisihle  by  p  (g  •I'.W),  Cor).     Transposing  the  terms  of  the 

last  niciuljer  of  the  equation  which  arc  not  divisible  by  y>,  we 

lind 

rt"  —  (fi  —  1)''  —  1  —  a  miibiple  of  p. 

or       <i''  —  ((  —  [{((  —  I)''  —  {<(  —  \)^^  =  a  multiiilc  of  />. 

Supposiiiij;  ./•  =  ",*,  this  equation  shows  that  '■.*''  — '^  is  a 
multiple  of  /*;  then,  supposiiii;  r  ■= ',].  we  show  hy  ij  '^'M, 
Th.  I!,  that   :)''  —  '.]  is  such,  a  multiple,  and  so  on,  indelinitely. 


II 


ence 


(I- 


—  It  =  a  multiple  (»!'/', 


whatever  he  a.  But  (0'  --  a  =  {iii'  ^  —  1)^/,  and  hecause  this 
l)roduct  is  divisible  by  />,  one  of  its  factors  must  be  so  divisible 
(>5  •,*;><!).     Hence,  if  a  is  prime  to^^,  uJ'  '  —  1   is  diviisiblc  hy  p. 


2o8 


coy  TL\  UED    FIL 1 CTIONIS. 


|l 


'\ 


1     t  I 


C  H  A  P  T  E  R     I  I . 

OF     CONTINUED      FRACTIONS. 

fil.*>.  Any  [irojici-  fnic-tioii  luiiy  l)e  ivjm'scnk'tl  in  tin.'  form 

-  -,  wlii'iv  ./•,  i.s  i,nT;ik'r  tiiuii  iuiii\',  but  is  not  iicfessnriiy  ii  u  lidjo 

•'t  ■  .  ■ 

luiiiilu'r.     ir.Vj  l)c  till.'  ^it'iifc'st  \\\\i)\v  lunnlu'i'  in  .i\,  wo  can  put 


ir, 


^'x  +  .. 


■svlu'iT  .'•„  will  be  <:'VuU'r  tliiin   unity.     In  Uie  siune  way  \vu 
may  put 


.r„  =  r/..  + 


1 
1 


^•;!   =  ^'3   +       ■» 
''  4 


fie 


itc. 


Tf  for  oach  ./■  wo  substilulc  its  expression,  the  IVaction 


"vvill  take  i\\v  l<-!-ni 

J_  _  1 


1  ,    1 


"  *  rr„  -j ,  ete.,  clc. 

jr  tlie  substitutions  are  continued    indelinitely,  the  foi'ni 
Avill  bo  1 

r^,  + p 

'<.  I  r 


<u  ■\- 


<'r. 


Such  an  c\pressii>n  is  calli'd  a  continni'd  fi-Mction, 

A/'.  \  Continued  Fraction  is  one  (d'  \\hi(di  tlii^ 
(IfMioiiiiiiator  is  ;i  whole  imiiilxr  j>lus  a  tViictioit ;  the 
(Icnoiiiiiiator  of  tills  last  IVactioii  a  vvliuk*  imnilx'r  i)lus 
11  liactioii,  etc. 


fut( 


k-Mc 


and 


CONTIXUED    FU.  1 CTTOXS. 


259 


form 
vhole 
n  init 


'ly 


wc 


A  coiitiiuicd  fniclloii  may  cHIkm-  icniiiiiatc  with  ono  of  its 
dt'iiuiiiiiiators  or  it  may  extend  iiulelinitely. 

Jh'f.     Wlicn  tlic  imnilxT  of  (|U()ti<'iits  a  is  iinitt*,  tl>o 
IVactiijii  is  said  to  be  Terminating. 

*^41.  PuoHLEM.     To  find   tlic    U((Iiic   of  a  roiidiiitnl 
fi'dcfiou. 

Wc  first  find  tiic  value  Aviieii  wc  stop  at  the  first  denomina- 
tor, then  at  the  second,  then  a'  the  third,  ete. 

Ucsing  only  two  denominators,  the  I'ruetion  will  be 

.r,  1         rrj./o  4-  1' 


a.  + 


.'"o 


/'  heinp^  ])ut  for  the  true  vahie  of  the  fraction. 

To   liiul    the  e\|)ression  with    three    terms,  we  put,  in   tin; 
preceding  expression,  r^_,  -|-    -  in  place  of  x.,.     This  gives 


; 


<on 


(ti  + 


I 


3  


o.,.r 


+  1 


lU 


('x'<t  4-  J  +  1 


{<U"-i    +   l)-'"3    -i- 


^^ 


To  find  the  result  with  the  fourth  denoniiiuitor,  we  snhsli- 


.r    fol'Ul 


Ik 

1)11 

Icr 


;  tho 
plus 


tutc    J-.,  r^il.,   + 


r  =. 


Tl 


le  II  iciioii  iieeomes; 


{"-. 


a. 


\)>\  -^ 


<i , 


[{<t\((«    +   i)«3    +   "l  I  •'■»    4-   "l"3    +    i 


(") 


To   iuvesti;^ate   the  gen(M'al   law   accui-ding    to   uhic1\    tlu^ 
successive  expressions  proceed,  we  put 

7',  the  encfVieient  of  .r  in  any  iiuinei-ali»r ; 

7*',  tlie  fiMthcieiii  of  ,/•  in  the  denfuninaloi-; 

^,>.  the  terms  not  multiplied  hy  ./■  in  the  niinu'ratnr  ; 

Q  y  the  terms  imt  multiplied  hy  ./•  in  the  den miinator ; 


.'II 


id  we  distinguish  the  vai'ious  cMpressions  hy  giving  each   /' 


und  Q  the  same  index  as  the  x  to  which  it  lielongs. 


2G0 


CO  NT  IN  UKD    FIL 1  ( '  TIONti. 


•) 


(*) 


Till  n  wc  m;ij  rcpr'.'seii'  each  value  of  Fm  the  form, 

^'  -  ^';^, +  <;>'/ 

>vlioro  /  may  iako  any  value  nooessary  to  (listitijruisli  the  frae- 
lion.     ("oiiii)ariiii;"  with  tlic  IVacliutis  as  written,  wo  see  that: 

l\  =0,  (2,  =  1,     l>\  =  i,  Q\  =0; 

y'„  =  1,  (>„  =r  0,     P;  =./,,  (/„  =1;        (r) 

To  show  tliat  tills  form  will  contiiuio,  how  far  soever  v,q 
carry  the  computation,  we  })ut  in  the  e.\i»ressioii  (//)  the  general 
value  of  .^1, 


which  jjives,         F  = 


xi  =  ni  -\-  -— , 


(rf) 


To  show  the  f^eneral  law  of  .succession  of  the  lerms,  let  us 
cnnip!ir(>  the  general  e([uation  {b)  with  {d).  Tutting  i+1  for 
/  in  (//),  it  heeomes, 


^_  Aj^ia*M  +  <?i  1 


If  1       it  1        '         ^»  ;  1 

Comparing  this  with  (^/),  we  find 

whence,  Qi  =  /*/-!• 

Sul)stituting  this  value  of  Qi  in  the  equation  previous,  it 

lieeoines 

Working  in  the  same  way  witii  the  denominators,  we  (ind 

/■;,,  =  «,/>;  +  /•;.,.  (y) 

])_)  suppobii'g  /  (o  take  in  su.  cession  (he  values  1,  !^,  3,  <'tc., 


lia 


P 


(^) 


fnic- 


(d 


neuer.'il 


('0 


:vioiis,  it 

(/) 
we  tiiul 


CONTINUED    FIL 1  CTIONiH. 


2(*1 


tho.«o  foniiulu^  sliow  that  the  successive  values  of  F  may  ho 
coinimtc'd  thus: 


1\  =  1,  \ 


(from  c) ; 


Also, 


1\  ^  a\l\  -V  l\z. 

1\  =  <'rJ\  +  /*4» 
etc.,  to  any  ex  tent. 

/>;  ==  1, 

etc.  etc. 

Siiioi'  each  value  of  Q  is  equal  to  the  vahio  of  P  haviup:  (ho 
next  snuiller  index,  it  is  not  necessary  to  conii>utc  the  Q'&  sep- 
arately. 

If  the  fraction  terminates  at  the  ><'*  value  of  a,  wc  shall 

have 

Xn  —  ((,1,  exactly. 

If  it  (loos  not  terminate,  we  have  to  ncijlcct  all  the  «lenom- 
inators  after  a  certain  point;  and  calling  the  last  denominator 
wo  use  the  h^,  we  must  suppose 

^'n   —  On. 

In  cither  ease,  the  expression  (//)  will  pive  the  value  of  the 
fraction  witii  which  we  sto})  hy  putrinir  '*  =  w  ami  Xn  =  (in. 


Therefore, 


F  ^ 


On  I'n  +  Q 


I  > 


or,  suhstituting  for  Q   and  Q'  their  values  in  (y), 


F^ 


ttnP'n  -k-   Pn-l 


But  the  general  expressions  (/)  and  {1/)  give 


«■■-*•■ 


m  t 


•I 


mm  ^' 


202 


LVNTIN UED    Fit.  1  (  TIOSS. 


'riu'i-cruro, 


f'n   l\l    4-     /'/i     1    =    Pith 

('u  I'll  4    ^■'/(-i  =  Z'/i-i. 
/Vi 

Tllrn  fort',  lit  jiiid  Ihr  nthic  nj'  llw  I'vurllou  Id  llw  n"*- 
Icrni.irr  Inin'  onlijln  nmifin/f  Ihc  nilucs  of  l'n\  ninl 
r„  1,  it'itliotil  lakiiii>  (fill/  ticnmiil  nj'  (J. 

KxAMi'Li;.     Tiikc  the  IVacdon, 


1 


1  -f- 


l 


:{  + 


I 


1 


a  etc. 


Hero,     n^   —  ],     ^'i  =  2>     f's  —  '•),....  fit  =  1. 

We  iu)\v  liMvc,  Iiy  conliiiiiin*;  the  r-.Timihi'  {')  iiiul  (/),  ainl 
iisini^'  (llu^■e  vjiliU'S  of  a^.  a.,,  .  tc. : 

J\  =  n, 
J\  =  1, 

I\    =   fl.J\^    -f     /'i      -    II.,    :^    •>, 

/'b  =  i,J\    I-  /'.  =  M  +  ;>  -  MK 
J\  =r  ./,/',  -H  /;  =  r>.;{M  -f.  7  =  lo7, 
olc.  cLc.  (.tc. 

y  ,    —   1, 

p;  =  rr, .  - 1, 

P\  -  uj'\  4    /'',  -  4.10  -f  :J  =  4.*], 

r\  —  aj>',  i.  r\  ~  r»-4;j  -f  lo  =  "225. 

Tlicicl'oiv.  <i!])p()sin^^  in  .sncci-ssioiK  n  =:  \,  w  ~  5,  v  ■= ',), 
vie,  we  liavL'  ior  the  .suoccssivo  ai»|tri)\iinate  values  of  llic 
fuiL'tioji, 


hr 


(I 


ml 


/■),  uikI 


COSTL\ L  i:i)    Fit. \CTWN8. 


2g:j 


/'., 

T'or  n  =  1, 

/,'    —      - 
■'  I  —  />' 

2 

=    1. 

Kor  //  =;  ;!, 

2 

I'or  tt  =z  5, 

-*   6    —     />' 

0 

-  ^''1 

TIr'So  succcssivo  apiJroxiniatc!  values  ol'  (lie  coiUimicd  IVac- 
tiou  are  calKd  Converging  Fractions,  ov  Convergents. 

•^l.!.  The  forms  {/)  and  (/y)  may  bo  expressed  in  words  as 
follows: 

T/ir  iiunivvdtor  af  citch  ronrrrtiPut  ii^  fornird  hi/  iniil- 
ti/)h/in^'  i/ir  prcrctlin'J  iiii inrrdtor  hij  tlw  corrcs/joiH/ini; 
a,  (tnd  (iddin'j  the  second  numeral 


l»' 


pi 


t)( 


liict. 


The  successive  denominators  are  formed  in  \\u\  same  way. 


KxAMPLK.     The  ratio  of  the  motions  of  the  snn  and  t)io.»ii 
relative  to  the  moon's  node  is  given  by  the  euntiuued  iVaclion: 


r^-i- 


1  i- 


Ji4- 


1  + 


-1  + 


l  

:{  -f-  I't- 


TiCt  ns  find  the  successive  converge!! Is.      W-'  [>iil  (li     df 


nominators  (f^  =^  i'i,  '/..  =  1,  etc..  in  a  linr,  thn,- 


a     = 


1  o 


P     =     0 


1 


;{ 


p'  =    r   i-y   la'   'ss' 


1  1!) 

»i'     )ii'Z 


r,\ 


« < 


0 


Under  tf.  we  write  llic  frac  iion   , ,  wliidi  i-  al\\av>  Ibo  one 


k   of  tlu* 


with  which  to  start,  because  P 


Next  to  the  ri'dil  i? 


<(, 


lieeau.se  7' 


0  and   /'',  :=  I   (S  -^4+-  r). 
„  =  1  and   /''.,  =:^  'I.     After 


this,  wo  multiply  each   term  by  flu'  multii)lier  </  above  il,  and 


204 


VOXTh\  UKIJ    FHA  CTIONS. 


adil  flio  torm  to  tlio   left   to  obtain   ilic  torm  on  tlie  riglit. 
'riiiis,     y-I  -f  1  =  ;j,     y.i:5  +  l;i  =  ;}«,     cK;. 

ICx.  2.     'I'»»  comp'.itc!  the  coiivcrgeiits  of 


2  4  -^-. 
4  + 


1 


a  + 


4  etf'. 

a     =     2,     4,     2,      4,      2,        4,       etc. 

NuincMMtors,        0      14       9       40  80 

_       .        ..  _  ^    -       etc 

Di'iioiniuiitors,     1'     2'     li'     20 '     8'j'  1U8' 


EXERCISES, 

Kodnce  the  following  coutinuctl  fractions  to  vulgar  frac- 


tions; 


I. 


3  + 


7  + 


IG 


2. 


3  + 


2  + 


3  + 


3  + 


1+-   -3 
3  +  - 


4- 


3  + 


a  4- 


5  + 


o; 


h  + 


1  +  .. 


If 


210.  PnoitT.KM.     To  express  a  fractional  quaniitij  as 
a  co/i tinned  fraction. 

Ijct  //  1)0  the  given  fraction,  less  than  unity.     Conipuie  j\ 

from  the  formula, 

1 

Let  rr,   1)0  the  whole  number  and  R'  the  fraction  of  .r,. 

Then  comjiute 

1 


CON  TIN  UED  Fit  A  VTIONS. 


2G.J 


frac- 


Tjot  ^8  bo  iho  wlidlc  nuinbcr  iiiid  //"  llio  frarlion  of  .Tg, 
We  continue  this  process  to  any  extt-nt,  iniK'ss  some  Viiliic 
of  a*  comes  out  u  whole  lumihcr,  wlion  we  stop. 


ExAMi'Li:.    Exiu'oss  ,,  -  as  a  coiitiuueil  rniclinn. 


-      1      _    ''^  _  9    ,    21 

""•  -  7^  -  2(>  -  ^^2U' 

1    _  20  _  5^ . 

II'  ~  ^l  "~      "^^1' 


.r.. 


_     1    _  21  _  1  . 

^3   —     f>"   —     r.     —  "^  ■+"  r  5 


R 

1 


^'*  -  7r-  1  -^' 

So  the  contiiuu'il  fraction  is 


.'.  a, 


—  ') 


.'.  a.,  =  1  ; 


•••  <'3  =  -i ; 


.-.  a,  —  5 ; 


A"  = 


A" 


21 
2()' 


21 
1 


^'     -5' 

A'"'    =r:  0. 


2  + 


1  + 


,       1 

o 


It  will  1)0  scon  that  the  process  is  the  same  as  that  of  find- 
ing the  greatest  common  divisor  of  two  numbers. 


+ 


(> 


i      Xy 


EXERCISES. 

Develop  the  following  (juotients  as  continued  fractions: 


I. 


113 
355" 


2. 


1041) 
332(i' 


(528 
i)25* 


247.  The  most  simple  continued  fraction  is  that  arising 
from  the  geometric  problem  of  cutting  a  line  in  extreme  and 
mean  ratio.     The  corresponding  numerical  problem  is: 

To  tliridr  itinhj  into  tiro  snrJi  fractions  thai  the  h'fis 
sJid/f  he  to  the  ^retttrv  ns  the  greater  is  to  uuitij. 

T.i't  r  bo  the  greater  fraction.  Then  1  —  r  will  be  the 
lesser  one.     W'v  ni'ist  then  have 


>  i\ 


l-r 


1, 


^ 


'» 


206 


(JONTINUEli    FUAVTIOys. 


wliicli  f]^iv('.s 

r»  =  1  —  r, 

or 

r'  -f  »•  =  • . 

or 

r(r+  1)  -.-.  1, 

I 

or 


I  +  /• 


Now,  IlL  us  put  for  /•  in  the  last  doiioiuinutor  the  o\])rc'Hsion 

■z -,  and  rei)out  the  procoss  indeHnitely.     Wf  shall  Iiuvo, 

I  +  r  ' 


r  = 


1  + 


1  + 


1 


1  + 


1 


1  etc.,  ad  ififinifum. 

Now  we  iniiy  form  the  sueeessive  converjreiUs  wliich 
iil>pro\iin:i(i'  to  the  true  vahie  hy  the  rule.  As  all  the  (lcii(»ni- 
inators  d  are  1.  uf  have  mo  n»ulliplyin<r,  hut  only  add  caeli 
term  to  (he  jireeeding  one  to  ol)lain  the  follow  iug  one.  Thus 
we  lind: 

0    112    rj    5     8     v.]    21     :u 

l'l'2'a'5'S'    1:5'    21'    ;M'    55'    ''^''• 
The  true  value  of  r  may  he  found  hy  solving  the  nuadratic, 

7'2  4-  7-  =  1 , 

-  1  ±  Vs 


which  jrives 


o 


The  positive  root,  with  wliich  alone  we  are  concerned,  is 


r  =  — Li"-^  -  o.(Ji8o;3;}i)0. 

2 


The  values  of  the  first  nine  convergents,  with  their  errors, 


are : 


1:1=  1.0, 


1  : 


0.5, 


error  =  +  0.382. 
«  —0.118. 


2:3=  O.OOG.... , 

« 

+  0.0480. 

3:5=  O.r.oo, 

<( 

—  0.0  ISO. 

5:8=  0.G25, 

u 

4-  O.OUG'Jr. 

( 'ON  TIN  Ui:i)    Fit  A  VTli)NS. 


*J07 


8  :  13  =  O.r.irK'JS.... , 

error  = 

—  o.oo-?r,r). 

13  :  21  =  O.ci'.M)!...., 

«< 

+  O.dOKH. 

j>i  :  34  =  o.ci ;»;»;...., 

u 

—  0.(MMi:;'.»r. 

31  :  'ui  —  U.Ol.SKS-^...., 

u 

4-  u.ooui  IS. 

t'Lc.                etc. 

etc. 

lirlatioiis  of  Siifcossivo  ('<»iiv<»rjj:<Mits. 

*2IS.   T  111:011  KM    I.      T/te    siicrrssii'o    coiircvjcnls    are 
nlhriniti'l ij  loo  hd'j^c  <in(l  too  snidll . 

Proitf.     Tlio   lir.-i   coiivergciiL   i.s        •     Tlio    true   (kiioin- 


(ti 


1 


iiuitor  Ik'Iii^  (I ^    f-       ,   llu'  (U'lioiuiiiiitor  a^  is  too  snmll,  iiiid 
tlicrdurc  tlie  I'mution  is  t(  o  liu'',a'. 

In  formiiif'  the  sccoiul  fnictinn,  wo  put         instead  of       • 

IJorauso  (t.,  <  .r^,  lliis  rractioii  is  too  largo,  which  makes  the 

1 

(1(  iiumiiKitor  (I.  -\-        too  siiiall. 

The  third  dcnoininator  a.^  is  too  small,  which  will  mako 
till'  |ii'o('odiiig  Olio  too  largo,  tho  iioxt  procoding  too  smalk  ;iiid 
Ko  on  altornati'lv. 

r\^  Ml'"'  I      '"        I  i  I  ■ 

JiiKoKKM   11.     //  (111(1       ,  he   (tiiii   tiro   ctjiiscciilLre 

^     n  a  ^ 

con  rr/'<Jr Ills,  Ihrii 

mil  —  )n' n  :^  ^  1. 

Proof.     Wo  show  : 

(k)  That  the  theorem  is  trne  of  tho  first  pair  of  convorgonts. 

(/i)  That  if  truo  of  any  pair,  it  will  ho  triio  of  the  pair  noxt 
following. 

(«)  The  first  pair  of  convcrgonts  are 


1 


a 


s 


«, 


,    1         a^(t.  +  1' 


which  gives  mu'  —  m  a  —  I,  tints  proving  («). 


n 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


1.0 


I.I 


I4S    — 

til 


M 

|||M 
IM 

1.8 


1.25      1.4      1.6 

< 

6"     

► 

Photographic 

Sciences 
Corporation 


23  WEST  MAIN  STREET 

WEBSTER,  NY.  14S80 

(716)  872-4503 


208 


CONTINUED   FRACTIONS. 


i) 


»l 


(/3)  Let 


m      m       m 


(1) 


n '      n  '      n 
be  three  consecutive  convergents,  in  which 

inn  —  iini  :=  ±  1. 
By  (/)  and  {(J)  wc  shall  have 

■}n'  =  am'  -\-  vi, 
■ii"  =  an'  +  n. 

;Mnltii)lying  tlie  second  equation  by  m'  and  subtracting  tlie 
product  of  the  lir^t  by  ?/,  we  have 


vi'u" 


1)1  )i   =z  in  n  —  mn 


wliich  is  the  negative  of  (1),  showing  that  the  result  is  T  1. 

The  theorem  being  true  of  the  lirst  and  second  fractions, 
must  therefore  be  true  of  the  second  and  third  ;  therefore  of 
the  third  and  fourth,  and  so  on  indefinitely. 

Corollaries.    Dividing  (1)  by  nn\  we  have 

m.       Ill  ,1         ^, 

;  =  +  — ,•       Hence, 

11        n  nn 

I.  Tlir  difference  hettreen  the  two  sitccessive  cojiverg- 
ejits  is  equal  to  luiitij  divided  hij  the  product  of  tke 
denoniintitoi's. 

Becau:^e  tlie  denominator  of  each  fraction  is  greater  than 
that  of  the  preceding  one,  we  conclude: 

II.  T/te  difference  heticccii  tiro  consecutive  converge nts 
coiistautlij  diminishes. 

Combining  these  conclusions  with  Th.  I,  wc  conclude  : 

III.  Each  rail  r  of  a  convergent  always  lies  hetween 
the  values  of  the  tiro  preeedini^  convergents. 

For  if  R^,  7?5,  7i*g  be  three  such  fractions,  and  if  B^  is 
greater  than  R^,  then  R^  will  be  less  than  R^.  But  it  must 
be  greater  tlian  R^,  else  we  should  not  have  R^  —  R^  numer- 
ically less  than  R^  —  R^.  Hence,  if  we  arrange  the  successive 
convergents  in  a  line  in  the  order  of  magnitude,  their  order 
will  be  as  follows: 

7?„  7?„,  R,.  ..../?„  7?„  R„ 

each  convergent  coming  nearer  a  true  central  value.     Hence, 


CON  TIN  UED    FliA  CTIONS. 


200 


t     (I 


(1) 


IV.  77ie  trite  value  of  the  coiitiiuocil  fraction  al- 
ways lies  between  the  values  of  two  consecutive  con- 
vevgents. 

Compuring  with  (I),  we  coiicludo  : 

V.  Tlie  error  which  we  make  hy  stopping  at  any  con- 
vergent can  never  he  greater  than  unity  divided  hy  the 
product  of  the  denominators  of  that  convergent  and  the 
one  next  following. 

EXAMPLE. 

Referring  to  the  tabic  of  values  of  j^{V6  —  1)  in  §  2-17, 
we  see  that : 


Error  of    2  :  3  <;-,-- ; 

3-0 

Error  of    3  :  5  <  ^  , ; 

0-8 

etc. 


(for  .048G  <  jX 
(for  .018    <  ly 


etc. 


Hence,  in  general,  continued  fractions  give  a  very  rapid 
approximation  to  the  trne  value  of  a  quantity.  Thei"  princi- 
pal use  arises  from  their  giving  approximate  values  of  iiv,  tional 
numbers  by  vulgar  fractions  Avitli  the  smallest  terms. 


EXAMPLE. 


Develop  the  fractional  part  of  V2  as  a  continued  fraction, 
and  lind  the  values  of  eight  convergents. 

Because  1  is  the  greatest  whole  number  in  \/2,  we  put 

V2  =  l  +  l;  (1) 


whence, 


X  = 


V2  —  1 

Ealionalizing  the  denominator,  §  185, 

a-  =  V2  +  1. 
Substituting  for  V^  its  value  in  (1), 

1 


X  =  2  -\- 


m 


H\ 


270 


CON  TIN  UED   FllA  CTIONS. 


Putting  this  value  of  a;  in  (1)  and  again  in  the  dcnominatorj 
and  repeating  the  substitution  iudellnitely,  we  find 

"i 


V^  =  1  + 


2  + 


2  + 


^  "^  2  etc. 


Forming  tlie  convcrgents,  we  find  them  to  be 

1     ?     A     1^     ??     J'^      V'l     i^i^ 
2'     5'     VZ'     29'     70'     IGU'    408'     'Jb5' 


etc. 


Adding  unity  to  each  of  them,  we  find  the  approximate 
vahies  of  ■v/'-i : 


3 


D 


17      41 


99 

70' 


239      577      1393 
T"G9'     408'      9b5  ' 


etc. 


liEM.  Tlic  square  root  of  2  may  be  employed  in  finding  a 
right  angle,  because  a  right  angle  (by  Geometry)  can  be  formed 
by  three  pieces  of  lengths  proportional  to  1,  1,  ^/^Z.  If  we 
make  the  lengths  12,  12,  17,  the  error  will,  by  Cor.  V,  bo  less 

than  T^^,  or  less  than  -—  of  the  whole  length. 

EXKRCISES. 

Develop  the  following  square  roots  as  continued  fractions, 
and  find  six  or  more  of  the  partial  fractions  approximating  to 
each  : 

I.    \/3.  2.    Vs.         3.    Vo.         4.    Vio. 

5.   Develop  a  root  of  the  quadratic  equation 

n'2  —  ax  —  1=0, 
commencing  the  operation  by  dividing  the  equation  by  x. 

l*oriodic  Coiitiiiuort  Fractions. 

249.  Dff.  A  Periodic  continued  fraction  is  on.?  in 
wlilcli  the  dcnoniiniitoi's  n^peat  themselves  in  regular 
order. 


^ 


CONTINUED    FRACTIONS. 


271 


Example.    A  continued  fraction  in  wliich  the  successive 
denominators  are 

2,    3,    5,    '2,    3,    5,    2,    3,    5,    etc.,    ad  infinitum, 
is  periodic. 

A  poi'iodic  continnod  fmction  can  be  expressed  as 
the  root  of  a  quadratic  equation. 

EXAMPLES. 


I. 


l-f 


2  + 


i  +  J- 


0 


+  etc. 
If  we  put  X  for  the  value  of  tliis  fraction,  we  have 

^  _  1 

^  '^  2T^" 
We  find  the  value  thus : 

1,        3  +  a?. 

1  2j\-x 

1 '  3  +  a;* 

Because  this  expression  is  x  itself  we  have 

—  ^  +  a; 
^~3T^' 

which  reduces  to  the  quadratic  equation 

x^  +  2x  =  2. 

2.  Let  us  take  the  fraction  of  which  the  successive  denom- 
inators are  2,  3,  5,  2   3,  5,  etc.,  namely, 

1 


X  = 


2  + 


3  + 


5  + 


2   +  7 


1 


3  +  etc., 


'I 


273 


or. 


CONTINUED   FllA  CTIONS. 


X  = 


3  + 


3  +  ^ 


Wc  compute  thus: 

3,  3, 

0  1 

1'        r 


b  -{•  X 


X  -{-  5. 
3 

r 


3x  4-  16 


7x  +  37 

Ilcncc  wc  have,  to  determine  x,  the  quadnitic  equation, 
3x  +  10 


X  = 


7.r  +  3r' 


or 


72-2  +  3-la:  =10. 


250.  Development  of  the  Root  of  a  Quadratic  Eqvation. 
A  root  of  a  qujidnitic  equation  may  be  developed  in  a  continued 
fraction  by  the  following  process.  Let  the  equation  in  its 
normal  form  be  (§  192), 

mx^  4-  nx  +  ^)  =  0,  (1) 

m,  11,  and  ^;  being  whole  numbers.     We  shall  then  have 

n  ±  V^'2  —  ^mp 


X  = 


2m. 


Let  a  be  the  greatest  wliole  number  in  x,  which  we  may 
find  either  by  trial  in  (1)  or  by  this  value  of  x.    Then  assume 

X  =  a  -\ , 

X, 

and  substitute  this  value  of  x  in  the  original  equation.  Then, 
regarding  x^  as  tbe  unknown  (juantity,  we  reduce  to  the  nor- 
mal form,  which  gives 

(ma^  +  na  +  p)x^^  +  (2ma  +  n)  x^  +  m  =  0.         (3) 
If  rtj  is  the  greatest  whole  number  in  a-j,  Ave  put 

1 

^1  =  ^1  +  !,-» 

and  by  substituting  this  value  of  x^  in  (2),  Ave  form  an  equa- 
tiv)n  in  Xo.  Continuing  the  transformations,  we  find  the 
greatest  whole  number  in  x„,  which  will  be  called  rtg,  and  so  on. 
The  root  Avill  then  be-  expressed  as  a  Avhole  number  a  plus 
the  continued  fraction  of  which  the  denominators  are  a^,  a.^, 


' 


(1) 


(3) 


. 


BOOK    X. 
THE     CO  MB  IN  A  TOR  V    A  NAL  J  \SIS. 


CHAPTER     I. 

PE  R  M  U  TATI  0  N  S. 

251.  Bef.  The  different  orders  in  wliicli  a  niimhor 
of  tilings  can  be  arrangc^d  are  called  their  Pormuta- 
tions. 

Examples.     The  pormntations  of  the  letters  a,  I,  are 

(lb,     ha. 
The  permutations  of  the  numbers  1,  3,  and  3  are 
Vn,     13;>,     313,     231,     312,     321. 

Problem.     To  find  how  many  permutations  of  any 
given  niunher  of  tilings  are  possible. 
Let  us  put 

Pi,  tlic  number  of  permutations  of  /  tilings. 

It  is  evident  from  tlie  (ir.st  of  the  above  examples  that  there 
are  two  permutations  of  two  things.     Hence, 

P    —  -1 

To  find  the  permutations  of  three  letters,  n,  h,  r,  we  form 
three  sets  of  permutations,  the  first  beginning  with  a,  the  sec- 
ond with  b,  and  the  third  with  c. 

In  each  set  the  first  letter  is  to  be  followed  by  all  possible 
permutations  of  tlie  remaining  letters,  namely: 

In   1st  set,    after    a    write    he,    ch,    making    ahr,    ach. 
-<1      '*        "       b       "       ar,   ca,         "  bar,     bra.    ' 


3d      " 
IS 


ab,   ba, 


i< 


cab,    cba. 


I  ! 


274 


PEliMUTATlONS. 


M 


T\y 


'IMie  jHTinutiitions  of  ;/  thiii^^s  fan  1)0  divided  into  sots. 
The  first  sot  bo^iiis  with  tho  first  tiling,  followod  by  ail  })ussi- 
l)lo  ponnututions  of  the  remaining  n  —  1  things,  of  whicli  the 
iiunibcr  is  l\  \.  Tho  second  set  begins  with  tlio  second  tiling, 
followod  by  all  possible  permutations  of  the  remaining  n  —  1 
things,  of  which  the  nnnii)er  is  also  P„  \,  and  so  with  all  ii 
sets.  Jlonco,  whatever  be  ;/,  there  m  ill  bo  n  sots  of  P^-i  per- 
mutations in  each  set.     Therefore, 

Pn  =  nPn-i. 
This  equation  enables  us  to  find  Pn  Avlioncver  we  Icnow 
P/i-h  iiii*^^  thus  to  form  all  possible  values  of  /'«,  as  follows: 

It  is  evident  that 
We  have  found 


Putting  71  =  4,  Ave  have 
n  =  5,   "     " 
etc. 


(< 


P,  =  1. 
P„  =  2-1  =  2! 
pI  =  ;3.2-l  rr  3!  -  0. 
p\  =  U\  =  4 !  =  24. 
Pg  =  5/%  =  5!  =  120. 
etc.  etc. 


It  is  evident  tliat  the  number  of  permutations  of  n  things 
is  equal  to  the  continued  product 

1  •  2  •  3  •  4  .  .  .  .  >?, 
■which  we  have  represented  bv  the  symbol  n !  so  that 

P'n  =  u\ 

EXERCISES.* 

1.  Write  all  the  permutations  of  tlie  following  letters  : 

bed,        ard,        ahil.        ahcd. 

2.  What  lU'oportion  of  the  i)ossil)lo  permutations  of  the 
letters  a,  c,  w,  i,  make  well-known  English  words? 

3.  Write  all  the  numbers  of  four  digits  each  of  which  can 
be  formed  by  i)erniuting  the  four  digits  1,  2,  3,  4. 

4.  llow  many  numljors  is  it  possible  to  form  by  permuting 
the  six  figures  1,  2,  3.  4,  o,  0. 

*  If  tlic  Ktudcnt  finds  any  difflculty  in  roasoiiincf  out  these  exercises, 
he  is  recommiMidcd  to  try  Hiuiiiar  caws  in  wliicli  few  symbols  are  involved 
by  actmdly  I'onuin.i?  tho  permutations,  uatil  he  clearly  sees  the  geiicial 
principles  involved. 


1 


(hi  11,1:, 

n  -  1 

ill!  Ji 

-1  I'^'i'- 


yeiiciul 


f 


PERMUTATIONS. 


2i:y 


I 


i 


5.  At  a  dinner  party  a  row  of  0  pliitcs  is  set  for  Uio  liost 
and  5  guosts.  In  how  many  ways  may  tlicy  be  seated,  Kuhject 
to  the  condition  that  the  host  must  have  Mr.  Brown  on  Ills 
right  and  Mr.  11l:  ^'Iton  on  his  left  ? 

6.  Of  all  nuniboi  that  can  be  formed  by  permuting  the 
seven  digits,  1,  JJ  ....  7: 

{(()  How  many  Avill  be  even  and  how  many  odd  ? 
{h)  In  how  many  will  the  seven  digits  be  alternately  even 
and  odd  ? 

(c)  In  how  many  will  the  three  oven  digits  all  be  together  ? 
{(I)  In  how  many  will  the  four  odd  digits  all  be  together? 

7.  In  how  many  permutations  of  the  8  letters,  a,  h,  c,  d,  e, 
f,(),h,  will  the  letters  (/,  c,  /',  stand  together  i;i  alphabetical 
order  ? 

8.  In  how  many  of  the  above  permutations  Vvill  the  word 
deaf  be  found  ? 

9.  In  how  many  of  the  permutations  of  the  first  9  letters 
will  the  Avords  affe  and  bid  be  both  found  ? 

10.  A  party  of  5  gentlemen  and  5  ladies  agree  with  a  math- 
ematician to  dance  a  set  for  every  way  in  which  he  can  divide 
them  .'nto  couples.     How  many  sets  can  he  make  them  dance? 

II  In  how  many  of  the  iiermutations  of  the  letters  a,  h,  c, 
d,  c.  will  d  and  no  other  letter  be  found  between  a  and  e. 

12.  In  how  many  of  the  permutations  of  the  six  symbols, 
r,  h,  c,  d,  c,t\  will  the  letters  (d)c  be  found  togetlier  in  one 
group,  and  the  letters  dcf  in  another? 

13.  How  many  permutations  of  the  seven  syml)ols,  a,  h,  c, 
T,  c,f\  (J,  are  possible,  subject  to  the  condition  that  some  per- 
mutation of  th(  letters  abc  must  come  first  ? 

14.  The  same  seven  symbols  being  taken,  how  many  per- 
mutations can  be  formed  in  which  the  letters  abc  shall  remain 


together  ? 


Perimi  tat  ions  of  Sots. 


252.  Dcf.  When  permutations  are  formed  of  only 
s  things  out  of  a  whole  number  71,  they  are  called  Per- 
mutations o-  n  things  taken  6^  at  a  time. 


ii 


l.-^.mmmrm'm^*  m 


270 


PKUMUTXTinys. 


ExAMPLK.  Tlic  porniu  tat  ions  of  the  three  letters  a,  b,  Cy 
taken  two  at  a  time,  are 

(lb,    bn,    nc,     en,     be,     ch. 

The  permutations  of  1,  3,  3,  4,  taken  two  at  a  time,  arc 

1;>,   i;j,  14,  -a,  A  ^4,  ;31,  Wl,  U,  41,  \i,  43. 

Pi{01jli:m.  To  find  the  naniOcr  of  pcrniutations  of 
n  t/tiii^s  taken  s  at  a  time. 

Sui)pose,  tirst,  that  we  take  two  things  at  a  time,  as  in  tho 
above  examples.  We  may  choose  any  one  of  the  n  things  as 
the  lirst  in  order.  Which  one  soever  we  take,  we  shall  have 
n  —  1  left,  any  one  of  which  may  be  taken  as  the  second  iu 
order.     Hence,  the  permutations  taken  "I  ut  ti  time  Avill  bo 

n  {ii  —  1). 

[Compare  with  the  last  e\'am])le,  where  7i  =  4.] 

To  form  the  i)ermutations  .3  at  a  time,  we  add  to  each  pcr- 

mutati(m  by  2*s  any  one  of  the  n  —  2  things  which  are  left. 

Hence,  the  number  of  permutations  3  things  at  a  time  is 

n{7i  —  1)  {n  —  2). 

In  general,  the  permutaticms  of  n  things  taken  s  at  a  time 
will  be  e([ual  to  the  continued  product  of  the  .v  factors, 

n  {ii  —  1)  {n  ~  2)  .  .  .  .  (n  —  s  +  1), 


which  is  equal  to  the  quotient 


si 


It  will  be  remarked  that  when  .s  =  ??,  we  shall  have  the 
case  already  considered  of  th(!  permutations  of  all  )i  things. 

EXERCISES. 

1.  Write  all  the  numbers  of  two  figures  each  which  can  be 
formed  from  the  four  digits,  3,  5,  7,  9. 

2.  Write  all  the  numbers  of  three  figures,  beginning  with 
1,  which  can  be  formed  from  the  five  digits,  1,  2,  3,  4,  5. 

3.  IIow  many  different  numbers  of  four  figures  each  can 
be  formed  with  the  digits  1,  2,  3,  4,  5,  G,  no  figure  being  re- 
peated in  any  number  ? 


J 


vEmnnwrioNs. 


2111 


arc 


i 


4.  P^xplain  liow  all  tlio  lumibors  in  (lio  ))ivr('(liii<,'  oxorcLse 
may  ho  wrillcii,  showing  how  nuuiy  miiuhuiti  k-^^iii  with  1, 
how  many  with  :?,  etc. 

5.  Ill  how  many  ways  can  3  gentlemen  select  tlieir  partners 
from  o  lailles? 

6.  How  many  oven  nunihcr.'^  of  15  (liiri.ront  digits  oaeh  can 
be  formed  from  the  seven  digits,  1,  ^,  ....  7  i* 

7.  How  many  of  these  numbers  will  consist  of  an  odd 
digit  between  two  even  ones  ? 


Circular  l*<'riiiiitatio!is. 

25.*>.  If  we  have  the  three  letters  a,  A,  c,  arranged  in  u 
circle,  as  in  the  adjoining  (igure,  then, 
liowevcr  we  arrange  them,  avo  shall  find 
them  in  alphabetical  order  1./ beginning 
with  a  and  reading  them  in  the  suitable 
direction,  llonco,  there  are  onlv  two 
different  circular  arrangements  of  three 
letters  instead  of  six,  namely,  the  two 
directions  in  which  they  may  be  in  al- 
phabetical order. 

Next  suppose  any  number  of  symbols,  say  a,  b,  c,  d,  e,  f,  fj, 
//,  and  let  there  be  an  equal  number  of  positions  around  the 
circle  in  which  they  may  be  placed.  These  positions  are  num- 
bered 1,  '2,  3,  4,  5,  0,  7,  8. 

For  every  arrangement  of  the  sym- 
bols w^e  may  turn  them  round  in  a  body 
Avithout  changing  the  arning(>ment. 
Each  symbol  Avill  then  pass  through  all 
eight  ]iositions  in  succession. 

By  performing  this  operation  with 
every  arrangement,  we  shall  have  all 
possible  permutations  of  the  eight  things 
among  the  eight  positions,  the  number 
of  which  is  8!,  which  are  therefore  eight  times  as  many  as 
the  circular  arranjremcnts. 


278 


vi:iiMVT.\riij.\H. 


•I 


lloncu  tlio  lUinilKT  of  ililRrciit  circular  urrun<^c'nu'nU  is 

8! 

--*,  wliii'li  is  the  siiiiit'  as  T!  . 

o 

Jii  ^jfciural,  il'  we  rcpivsciit  (iio  nunj1)er  of  circular  arrangc- 

liu'iit.s  uf  li  lliiM<;s  l»y  ('„,  wo  sslmll  luivo 

Tho  sjimc  result  nuiybo  reaclied  by  the  followiugrciisonini;'. 
'!!'()  I'onn  ;i  circular  urraiigeint'Ml,  \vc  uuiy  take  auy  ouo  .syiul)ol, 
a  for  exaiujdc,  put  it  iuto  a  llxed  positiuu,  .siy  (1),  aud  leave  it 
there. 

All  possible  arrangemenls  of  Ibo  syuibols  will  llien  be 
foriued  by  ])eruuiliug  tho  reuuiiuiug  syud)ols  auu)Ug  the  re- 
uiaiiiiug  pitsitious.      Ilciice, 

C,^  =  y',_i  =:  (y/-l)! 
as  before. 

EXERCISES. 

1.  lu  bow  nuiny  orders  can  a  l>arty  of  7  persous  take  their 
places  at  a  rouud  table? 

2.  Ill  bow  many  orders  can  a  bost  and  7  guests  sit  at  a 
round  table  in  order  that  the  host  may  have  tlie  guest  of  high- 
est rank  upou  his  right  and  the  next  in  rank  on  bis  left? 

3.  Five  works  of  four  volumes  each  are  to  be  arranged  on 
a  circular  shelf.  JIow  many  arrangements  are  possible  which 
will  keep  the  volumes  of  each  set  together  and  in  proper  order, 
it  being  indiU'erent  in  which  direction  the  numbers  of  the 
volumes  read. 

4.  In  how  many  circular  arrangemcnls  of  the  5  letters  a,  b, 
c,  (I,  (',  will  a  stand  between  d  and  d'f 

5.  If  the  10  digits  arc  to  be  arranged  in  a  circle,  in  how 
many  Avays  can  it  be  done,  subject  to  the  condition  that  even 
and  odd  digits  must  alternate?     (Note  that  0  is  even.) 

6.  The  same  thing  being  sup])osed,  how  many  arrange- 
ments are  possible,  subject  to  the  condition  that  the  even  digits 
must  be  all  together  ? 

7.  In  bow  many  circular  arrangements  ot  the  first  six  let- 
ters will  the  word  deaf  he  found?  What  Avill  be  the  differenco 
of  the  results  if  you  are  allowed  to  spell  it  in  either  direction? 


i 


\ 


t 


1'/:/:mltati(l\s. 


i>7D 


i 


VvvniiiiniUiUH  \v1kmi   S<»voral  of  tli«.»  Tliiiij^fs  ai*(^ 

Idciitu'al. 

?ii>l.  If  tlu!  same  lliiii;;  ai)i»i'tir.s  sovcnil  times  jiinoiif^  flio 
tliiii;i;s  t(j  bo  luTiuiitc'd,  i1j(j  miiiil)rr  ol' ilitTcrciit  itcnnutiilions 
will  l)L'  less  than  when  the  thill,^^s  are  all  dinViviit. 

KXAMi'Li;.     The  penniitatioiis  of  (Ki/j//  are 

(iit/jb,     (dnth,     ((hlid,     bddh,     b((ba,     bbud,  (1) 

wliieli  are  only  six  in  nimiher. 

I'U()ULI:m.  To  jlud  llic  innnhri-  of  pci'iniiliiliuiin  whcti 
sri'irol  of  I  he  lhiiii>x  ore  ii/ci/h'ro/. 

Lel<  us  lirst  examine  how  all  ^'1  i)enmitations  of  I  Ihinfrs 
may  he  lorincd  i'loin  I  hi-  uhove  0  |ieriuutations  of  (fobb,  Ia-1 
ns  (lislin,Lruish  the  two  ^/'s  and  I  Ik-  two  //s  Ity  aeeentint;  one  ol" 
each.  Then,  from  eaeh  permutation  as  written,  four  may  he 
formed  by  i)ei'mutin^  the  sinnlai'  letters  amon,i(  tlieniselves. 
For  example,  taking  ((bb(f,  nud  writin.^  ii  abb'a',  we  Iiave,  1)y 
permuting  the  similar  letters, 

ab'ba',     a'b'bn.     abb'a\     a'bh'a.  (-2) 

Tn  the  same  way  four  permutations,  differing  oidy  in  the 
arrangvment  of  the  aeeents,  may  he  formed  from  eaeh  of  the 
C  permutations  (1),  making  'l\  in  all.  as  there  ouyht  to  be. 

Cieueralixing  the  preceding  operation,  we  reach  the  follow- 
ing solution  of  our  [)robleni.  Let  the  symbols  to  be  permuted 
be  a,  b,  c,  ete. 

Suppose  that  a  is  repeated  r  times, 

it  k.  /l      "  "  V  " 

etc.  etc.  etc. 

and  let  the  whole  number  of  symbols,  counting  repetitions,  bo 
')i,  so  that 

w  =  r  +  •*•'  +  /  +  c'tc. 

[In  the  preceding  examjtlo  (1),  n  ~  4,  r  =  2.  s  =  2.] 
Also  put  A'„,  the  required  number  of  different  permutations 
of  the  n  synd)ols. 


280 


rERMUTATIONS. 


Suppose  tliesG  X,i  different  permutations  all  written  out, 
and  siipi)ose  the  symbols  which  are  repeated  to  be  distinguished 
by  accents.     Then : 

From  each  of  the  X,i  permutations  may  be  formed  Fr  =  rl 
permutations  i)y  permuting  th.e  (is  among  themselves,  as  iu 
(2).     We  shall  then  have  r  I  A\  permutations. 

From  each  of  the  latter  may  be  formed  *•!  permutations  by 
permuting  the  b's  amovg  themselves.  AVe  siiall  then  have 
.si  )'l  X  A\  ])ermutations. 

From  each  of  these  may  be  found  / !  iiermutations  by  in- 
terolianging  the  c's  among  themselves. 

Proceeding  in  the  same  way,  we  shall  have 

X'n  X  ?•!  X  6-1  X  /!  X  eic. 

possible  permntations  of  all  )i  things.     But  this  number  has 
been  shown  to  be  ;;  I     Thcreibro, 

Xn  X  rl  X  s\  X  t\  X  etc.  =  n\ 


By  division, 


Xn  = 


nl 


{-) 


rl  6-!  tl  etc' 

which  is  the  required  expression. 

"We  remark  that  if  any  syml)ols  are  not  repeated,  the  for- 
mula (3)  will  still  he  true  by  supposing  the  number  correspond- 
ing to  r,  s,  or  /  to  be  ) . 


['.M  tj 


f'^^ll 


III! 


EXAMPLES. 

The  number  of  possil)le  permutations  oi  aahb  are 
4!  U 


G,  as  already  found. 


2.  Tiie  possil)le  permutations  of  aaahbcd  are 

5040 


T! 


3!  -v. 


G-2 


420. 


EXERCISES. 

Write  all  the  permutations  of  the  letters: 
I.     aaah.  2.     aabc.  3.     aaabc. 

4.  IL)w  many  different  numbers  of  seven  digits  each  can 
be  formed  by  permuting  the  figures  11122;25  ? 


PERMUTA  TIOXS. 


281 


5.  If  every  different  permutation  of  letters  made  a  word, 
how  many  words  of  13  letters  caeh  could  be  formed  from  the 
word  jiassachusctts. 


i:^) 


\ 


The  Two  Classos  of     x^'imitations. 

255t  The  u\  possible  permutations  of  n  things  a''o  divisi- 
ble into  two  classes,  commonly  distinguished  as  even  i)ermu- 
tations  and  odd  permutations  in  the  following  way: 

We  suppose  the  n  things  first  arranged  in  alphabetical  or 
numerical  order, 

a,  h,  c,  d,  .  .  .  .        or        1,  2,  3,  4,  ...  .  n, 

and  we  call  this  arrangement  an  even  permiitatio)i. 

Then,  having  any  other  permutation,  we  count  for  each 
thing  how  many  other  things  of  lower  order  come  after  it,  and 
take  the  sum. 

If  this  sum  is  even,  the  permutation  is  an  even  one ;  if  odd, 
an  odd  one. 

EXAMPLES. 

I.  Consider  the  permutation  2051-43. 

Here  3  is  followed  by  1  number  of  lower  order,  namely,  1. 
''     G     *'        ''      "  4      ''        "         "  "       0,1,4,3. 

"     5     *'         "         3       *'         "         "  «       1,4,3. 

i(      -\       ((  a  A  a  <e  << 


i(     4-     "         "         1       "         "  "  **       3 

Then  1  +  4  +  3  +  0  +  1  =  9.     Hence  the  permutation  is  odd. 
2.  Consider  cdbea. 

Here  c  is  followed  by  2  letters  before  it  in  order,  namely,  la. 

"    d    "         "  2  "          "            "               "        ha. 

"     h     "         "  1  "          "            "               "       a. 

ii     c     '<          *'  1  *'          ''            "                **        a, 

Tlien  2  +  2  +  1  +  1  =0.     TlrMtce  the  permutation  is  even. 

Dcf.  Tho  total  number  of  times  wliicli  a  thing  loss 
in  order  follows  one  greater  in  oi-der  is  called  tlie 
Number  of  Inversions  in  a  permutation. 


ii'i 


m\ 


282 


PEUMUTA  TI0X8. 


ill  > 


^lllf! 


i| 


Example.  In  the  i)rcceiling  permutation,  2G5143,  the 
number  of  inversions  is  'J.     In  cdbed  it  is  G. 

Eem.  It  will  be  seen  that  the  class  of  a  permutation  is 
even  or  odd,  according  as  the  number  of  inversions  is  even  or 
odd. 

Theorem  I.  //,  in  a  pmnntdtion,  tiro  tJii/i^s  (tra 
intcrc]iaiii>cd,  the  class  will  be  clLuii^ed  fruni  even  to  odd, 
or  from  odd  to  even. 

Froof.  Consider  first  the  case  in  which  a  pair  of  adjoining 
things  are  interchanged.     Let  us  call : 

ik,  the  two  things  interchanged. 

yl,  the  collection  of  things  Avhich  precede  i  and  h. 

C,  the  collection  of  things  which  follow  them. 

The  first  jiermutation  will  then  he 

AihC*  {a) 

After  interchanging  /  and  k,  it  will  be 

AMC.  {!>) 

Because  the  order  of  things  in  A  remains  undisturbed,  each 
thing  in  A  is  followed  by  the  same  t hi. igs  as  before.  In  the 
same  way,  each  thing  in  C  is  preceded  by  the  same  things  as 
before. 

Hence,  the  number  of  times  that  each  thing  in  A  or  C  is 
followed  by  a  thing  less  in  order  remains  unchanged,  and, 
leaving  out  the  pair  of  things,  /,  k,  the  number  of  inversions 
is  unchanged. 

But.  l)y  interchanging  z' and  k,  the  new  inversion  ki  is  in- 
troduced. Therefore  the  number  of  inversions  is  increased 
by  1. 

*  This  fomi  of  al^rbraic  imtation  differs  from  those  already  used  in 
that  the  symbols  vl  -,\\u\('  do  not  stand  for  (piantities,  but  mere  collec- 
ti(ms  of  letters.  It  is  an  applieation  of  the  general  principle  that  a  single 
symbol  may  be  nsed  to  represent  any  set  of  symbols,  but  must  represent 
the  same  set  throughout  the  same  (lUfstimi.  A  and  (J  are  here  used  to 
show  to  the  eye  that  in  forming  the  jxrmutations  of  {h)  from  (/7),  all  the 
letters  on  each  side  of  ik,  preserve  their  relative  positions  unchanged. 


f 


! 


I 


ih 


ri'JIi  MUTATIONS. 


283 


.43,   the 

tatioM  is 
C'vc'ii  or 

i^s   (tre 

.  to  0(1(1, 

djoining 


)cd,  cac'li 
111  the 


lings  as 


or  C  is 
id,  and, 
versions 

?•/  is  in- 
icrcascd 


y  usod  in 
re  rollcc- 
t  a  win<rle 
reprosent 
'  used  to 
).  iill  the 
i-fd. 


4 


If  the  first  arrangement  is  ki,  tliis  one  iu version  is  removed. 
Hence,  in  either  case  the  number  of  inversions  is  changed  by 
1,  and  is  therelbrc  changed  from  odd  to  even,  or  vice  versa. 

lUudration.     In  the  permutation 

2G5143, 

tlic  inversions,  as  already  found,  are  the  follov.-ing  nine  : 

21,     Go,     Gl,     G-1,     G3,     51,     54,     53,     43. 

Let  us  now  interchange  5  and  1,  making  the  permutation 

2G1543. 

The  inversions  now  are 

21,     Gl,     G5,     G4,     G3,     54,     53,     43, 

the  same  as  before,  except  that  51  has  been  removed. 

Kext  consider  the  case  in  which  the  tl)ings  interchanged 
do  not  adjoin  eacli  other.    Suppose  tliat  in  the  permutalion 

h  a  (I  e  h  c  f 

we  interchange  a  and  /^  "We  may  do  this  by  successively  in- 
terchanging a  witli  d,  with  c,  and  Avith  li,  making  three  inter- 
changes, producing 

h  (I  e  h  a  c  f .  * 

Then  we  interchange  h  Avith  e  and  Avith  d,  making  two 
interclianges,  and  producing 

h  h  d  c  a  c  f , 

which  eflfccts  the  required  interchange  of  a  Avith  li. 

The  number  of  ihc  neighboring  interchanges  is  3-f  2  =:  5, 
an  odd  number.  Because  the  number  of  inversions  is  changed 
from  odd  to  even  this  same  odd  number  of  times,  it  Avill  end 
in  tlie  opposite  class  Avith  Avliich  it  commenced. 

TiiEOUE.Ai  IT.  Tlir  jms.vhJe  pcvnnddtions  of  ii  tldiigs 
are  one-half  crcii  and  onc-haTf  odd. 

Proof.     Write  the  v  !  ])(issil)le  permutations  of  the  n  tilings. 

Then  interchange  some  one  pair  of  things  [e.fj.,  the  tirst 
tAvo  things)  in  each  permutation.  AVe  shall  have  the  same 
permutations  as  before,  only  diil'ereiitly  arrang'jd. 


^    I 


m 


'li 


284 


PERMUTATIONS. 


By  the  change,  every  even  permutation  will  be  changed  to 
odd,  and  every  odd  one  to  even. 

Because  every  odd  one  thus  corresponds  to  an  even  one, 
and  vice  versa,  their  numbers  must  be  e<[ual. 

lUustrafion.  The  permutations  in  tlie  second  column  fol- 
lowing are  formed  from  those  in  the  tirst  by  interchanging  the 
first  two  ligures  : 

even,  2  1  3 

odd,  3  12 

odd,  12  3 

even,  3  2  1 

even,  13  2 


12  3 

1  3  2 

2  13 

2  3  1 

3  12 
3  2  1 


odd. 


')    Q     1 


odd. 

even. 

even. 

odd. 

odd. 

even. 


EXERCISES. 

Count  the  number  of  inversions  in  each  of  the  following 
permutations : 

T.     hcilagcf.  2.     hcarjdef.  3.     325941. 

4.     5432.  5.     82917304.  6.     829:i3G4. 

'ZoG,  Symmetric  Functions.  An  important  ap]»lication  of 
the  laws  of  permutation  occurs  in  the  problem,  hoAv  many 
different  values  a  function  may  acquire  by  permuting  the 
letters  which  enter  into  it.  We  readily  find  that  the  ex- 
pression (fitjc  takes  only  the  three  values  (i^bc,  l?ac.  and  c^db  by 
permutation.  Other  expressions  do  not  change  at  all  by  per- 
muting their  symbols. 

Def.  A  Symmetric  Function  is  one  wliicli  is  not 
changed  by  permuting  the  symbols  which  enter  into  it. 

EXERCISES. 

Show  that  the  following  functions  are  symmetric  : 
\.     a  -\-  h  -\-  c.  2.     ahc. 

4.     a-  {h  -  c)  -f-  Ij^  {c  —  a)  +  t'^  [a  —  b). 


^ 


1 


(JOMIUNATIONS. 


285 


langcd  to 


even  one, 

Inmn  fol- 
ngiiig  the 


I  following 

941. 
:13G4. 

•lication  of 
lioAv  many 
lilting  the 
at  tlio  ex- 
ind  c^ah  V)y 
all  by  per- 

ich  is  not 
ter  into  it. 


c : 


H 


CHAPTER     II. 

COMBINATIONS. 

2.>7.  Def.  The  number  of  ways  in  which  it  is  pos- 
sible to  selfct  a  set  of  .v  thinj^s  out  of  a  coHeetion  of  n 
thiuirs  is  called  tlie  Number  of  Combinations  of  s 
things  in  a. 

Ex.  I.  From  the  three  symboLs  a,  l,  c,  may  be  formed  the 

coui)lets, 

ah,         ac,         be. 

TIence  there  are  three  combinations  of  2  things  in  .3. 

Ex.  2.  From  a  stud  of  four  horses  may  be  formed  six  dif- 

feivnt  span.     If  we  call  the  horses  A,  B,  C,  D,  tlie  difll-rent 

s})an  will  lie 

AB,     AC,     AD,     BO,     BD,     CD. 

Rem.  1.  A  set  is  regarded  as  different  wlien  any  one  of  its 
separate  things  is  different. 

Rem.  '2.  Combinations  differ  fro.m  permutations  in  that, 
in  forming  a  combination,  no  account  is  taken  of  the  order  of 
arrangement  of  things  in  a  set.  For  instance,  ab  and  ba  are 
the  same  combination.  Hence,  we  may  always  suppose  the 
letters  or  numbers  of  a  combination  to  be  written  in  alpha- 
betical or  numerical  order. 

Notation.  The  numljer  of  combinations  of  s  things  in  n 
is  sometimes  designated  by  the  symbol, 

Problem.  To  jind  the  iiuinhcr  of  coinhinatwns  of  s 
thinos  ill  n. 

If  we  form  every  possible  set  oT  .v  things  out  of  n  things, 
and  then  |XTmiite  the  ■<?  tilings  of  each  set  in  every  possililo 
way,  we  sluill  have  all  the  permutations  of  n  things  talvcn  s  at 
a  time  (§  25-2).     That  is, 


Ji' 


•hi 


280 


COMBINATIUNH. 


'I 


■I:   !i 


express  the  nnmbor  of  pcrmuhitioiis  of  n  tilings  taken  s  at  a 
time.     But  we  liavo  i'oiuid  this  number  to  be 

n  {u  —  l){)i  —  '2)....{n  —  s-\-\ ). 

We  have  also  found 

I\  =.s\  =  1.2.3.4 s. 

llenee,     Cf  xsl  =  n  {n  —  1)  (w  —  2) {n  —  s-\-l), 

^n  _  n{n  —  \){7i  —  2)  ....(»  —  .9  +  f) 

=  {'^)  (§  228,  3) ; 


and 


or 


/-in 


which  is  the  required  expression. 

Hem.  For  every  coiii])ination  of  .9  things  wliicli  we 
can  talvi^  from  7i  things,  a  cooilbination  <)f  it,—  s  things 
will  be  left. 

ence,  Cl.   =  C  us- 

This  formula:  mr  v  be  readily  derived  from  the  expression 
for  the  number  of  combinations.     For,  if  we  take  the  equation 

pn  _  w! 

^'    -  s\  {u-s)l' 

this  formula  remains  unaltered  when  we  substitute  n  —  s  for 
s,  and  therefore  also  represents  the  combinations  of  u  —  s 
things  in  n. 

Def.  T\vo  combinations  wdiicli  together  contain  all 
the  things  to  be  combined  are  called  two  Complement- 
ary combinations. 

EXERCISES. 

1.  Write  all  combinations  of  two  symbols  in  the  live  sym- 
bols, a,  b,  c,  d,  e. 

2.  AVrite  all  combinations  of  three  symbols  in  the  same 
letters,  and  show  why  the  number  is  the  same  as  in  Ex.  i. 


i 


u; 


1 


Dii  s  at  a 


rliicli  we 
s  things 


eqiuiiiuii 


■ 

4 


co:JBiNATioys. 


287 


.1 

t 


3.  A.  span  of  liorsos  hoiiig  (lilTeiTut  Avlion  cilhor  linrso  is 
(•liaii;;e(i,  how  niaiiy  (I'l'Vivnt  span  nuiy  bo  formed  IVuui  a  stnd 

ot';j:-'    or  7?   oro? 

4.  If  fonr  [loints  aiv  niiirkcd  on  a  piofc  of  paper,  liow  many 
distinct  lines  can  bo  formed  by  joining'  them,  two  and  two? 
J  Tow  many  in  tlio  case  ^^^  n  points? 

From  each  ono  of  the  points  can  1)0  drawn  n  —  1  lines  to 
other  ])oints;  tlien  why  are  there  not  n  [n  —  1)  lines? 

5.  If  five  lines,  no  two  of  which  are  parallel,  intersect  each 
other,  how  many  points  of  intersection  will  there  be?  lIow 
many  in  the  case  of  n  lines  ? 

6.  If  n  straight  lines  all  intersect  each  other,  how  many 
diiferent  triangles  can  be  found  in  the  figure? 

7.  In  how  many  different  ways  may  a  set  of  four  things  be 
divided  into  two  pairs  ? 

8.  In  how  many  ways  can  a  party  of  four  form  jmrtnors  at 
whist? 

9.  In  how  many  ways  can  the  following  numbers  be  thrown 
with  three  dice  ■ 

{'<)     1,1,1;  {h)     1,2,2;  (.)    1,2,3. 

10.  A  school  of  15  young  ladies  have  the  privilege  of  send- 
ing a  party  of  5  every  day  to  a  picture  gallery,  j)rovided  they 
do  not  send  the  same  party  twice.  How  many  visits  can  they 
make? 


—  s  for 
of  n  —  s 

itrJn  all 
lement- 


tlve  sym- 
ihe  same 


^x.  I. 


Coiiibiiiatioiis  with  Repetition. 

358.  Sometimes  combinations  are  formed  Avith  the  liberty 
to  repeat  the  same  symbol  as  often  as  we  please  in  any  set. 

Example.     From  the  three  things  a,  l,  c,  are  formed  the 
six  combinations  of  two  things  with  repetition, 

aa,        ab,        ac,        hh,        he,        cc. 

Problem.     To  find  i.ie  iiiinihrr  of  coinhiuations  of  s 
tilings  in  n,  when  repetition  is  (dloiued. 

Solution.     Let  the  n  things  be  the  first  n  numbers, 


P 


t 


2SS 


COMBINATIONS. 


Form  all  possililc  sots  of  a  of  thcso  iinnibors  willi  iv])rtiiioii, 
till'  iiiiml  era  of  each  sot  being  arraiigrd  in  iium"rieal  onl'-r. 

Let  lis  1)0  the  re'|tiiro(l  lumiher  of  sets.     'IMien,  in  eaeli  set, 

Let  llie  iirsi  nninl)er  stand  nnelianged. 
Licreaye  tlio  'M  number  \>y  1. 

*'    3d       *'         "  ^\ 

"    4tli     ''         ''  3. 


it 
tt 


tt 


Jfi 


(( 


(( 


"We  shall  then  liavc  Jia  sets  of  s  numbers,  each  "svithont  rep- 
etition. 

Example.    From  the  numbers  1,  2,  3  aro  formed  with  repctitioii, 

11,  13,     13,     22,    2'3,    33. 
Tlu'ii,  increasiiifr  the  second  numbers  by  1,  we  have 

12,  13,    14,    23,    24,    34. 

The  greatest  possible  number  in  any  set  after  the  inerease 
will  be  n -\- s — 1,  because  the  greatest  number  from  whieh 
the  selection  is  made  is  n,  and  the  greatest  quantity  added  is 
s  —  1.  Hence  all  the  new  sols  will  consist  of  combinations  of 
s  numbers  each  from  the  u  -}-  s  —  1  numbers, 

1,  2,  3,  4,  .  .  .  .  w  .  .  .  .  ;i  +  s  —  1.  (a) 

No  two  of  these  combinations  can  be  the  same,  because  then 
two  of  the  original  combinations  would  have  to  be  the  same. 
Hence  the  new  sets  are  all  ditferent  combinations  of  s  numbers 
from  the  n  -{-  s  —  1  numbers  {(().  Therefore  the  number  of 
combinations  cannot  exceed  the  quantity  Cf. 

Conversely,  if  we  talvo  all  possible  combinations  of  s  differ- 
ent numbers  in  n  -{-  s  —  1,  arrange  each  in  numerical  order, 
and  subtract  1  from  tiic  second,  2  from  the  third,  etc.,  we 
shall  have  diiferent  combinations  from  the  iirst  7i  numljers 
with  repetitions.  Hence  the  number  of  combirUions  in  the 
second  class  cannot  exceed  those  of  the  first  class. 

Hence  we  conclude  that  the  number  of  coml)inations  oi  s 
things  in  7i  with  repetition  is  the  same  as  the  combinations  of 
s  things  in  n  -\-  s  —  1  without  repetition,  or 


(lOMlUNA  TTONS. 


280 


pciition, 
L'tU'li  set, 


m  = 


I'D 


liout  rep- 


letitioii, 


e  inevc:ise 
om  which 
uddcd  is 
initio  us  of 

Icause  then 
the  same, 
munhers 
:nimher  of 

If  s  cliffor- 
lical  order, 
\\,  etc.,  Ave 
iiumhors 
)iis  in  the 

itions  01  s 
Inations  of 


n  {u  +  1)  (u  -\-  2) (;/  +  s 


-1) 


1  •  ^  •  3  •  4  .  .  .  .  n 


EXERCISES. 

1.  Write  all  possible  combinations  of  3  nnm])ers  witli  repe- 
tition out  of  the  three  numbers  1, 2, 3 ;  then  inercaso  tlie  second 
of  each  combination  by  1  und  the  third  by  2,  and  show  that 
we  have  all  the  combinations  of  three  different  numbers  out  of 
1,  2,  3,  4,  5. 

2.  IIow  many  combi  ations  of  4  things  in  4  with  repeti- 
tion?    Of  n  things  in  «  ? 

In  the  last  question  and  in  the  following,  reduce  the  result  to  its 
lowest  terms. 

3.  IIow  many  combinations  of  n-\-l  things  in  n  —  1  with 
repetition  ? 

Special  Cases  of  Combi nations. 
259.    It  is  plain  that 

because  each  of  tliesc  combinations  consist  simply  of  one  of  the 
n  things.     Hence,  also, 

Ol^i  =  n, 

because  in  every  such  combination  one  letter  is  omitted. 
It  is  also  plain  that 

rin  _  -I 

because  the  only  combination  of  n  letters  is  that  comprising 
ttie  71  letters  themselves.    Hence  we  write,  by  analogy, 

although  a  combination  of  nothing  does  not  fall  within  the 
original  definition  of  a  combination. 

3(50.  The  formula)  of  combinations  sometimes  enable  ns 
to  discover  curious  relations  of  numbers, 

1.  Let  us  incpiire  bow  wc  may  form  the  combinations  of 


200 


COMUINATIONS. 


i 


'I 


.9-1-1  things  when  wo  have  those  of  c   tliiiicfs.     Let   tho  n 

things  from  which  the  combinations  are  to  he  formed  be  tho 

letters 

a,  b,  c,  (I,  (',/,  (j,  etc {n  in  numi)er). 

Let  all  tho  combinations  of  .9  4-^  "  "»eso  n  letters  bo  writ- 
ten in  alphabetical  onhT.     Then: 

1.  In  the  combinations  beginning  with  u,  tho  letter  <(  will 
be  followed  by  all  possible  combinations  of  .v  letters  ont  of  the 
n  —  1  letters  b,  c,  d,  etc.,  of  which  the  nnmber  is  ^'?~\ 

2.  In  the  coml)inations  beginning  with  b,  the  letter  b  is 
followed  by  all  combinations  of  «  letters  out  of  tho  n  —  2  let- 
ters c,  d,  e,  f,  etc.  Therefore  there  are  6'"  combinations 
beginning  with  b. 

3.  In  tho  same  way  it  may  be  shown  that  there  are  Cg 
combinations  beginning  with  c,   CT    beginning  Avith  d,  etc. 
The  series  will  terminate  Avith  a  single  combination  of  the  last 
.s'-Hl  letters. 

Since  we  thus  have  all  combinations  of  .*.•-{- 1  letters,  wo 
find,  by  summing  up  those  beginning  with  the  several  letters 
a,  b,  c,  etc., 

Ca     +  Ck     -\-  Cs     +  .  . . .  -I-  C,.  =  r ,.  1 1.  (a) 

Substituting  for  tho  combinations  their  values,  we  find 

By  the  notation  (§  228,  .3),  all  the  terms  of  the  first  member 
have  the  common  denominator  s\,  while  the  numerators  are 
each  composed  of  the  factors  of  s  consecutive  numbers.  Mul- 
tiplying both  sides  by  s!  and  reversing  the  order  of  terms  iu 
the  first  member,  we  have 

1.2-3 5  4-  2-3-4 s  +  1  4-  etc. 

etc.  etc. 

-[-  {n  —  s  —  V) .  .  .  .  {n  —  'd)  (n  —  2) 
-\-  {n  —  s)  .  .  .  .  {71  —  2)  {n  —  1) 

_  (w  —  .s)  ....(«  —  2)  (n  —  l)n 


tho  n 
be  tlio 


1)0  writ- 
er (t  will 

it  of  the 
-1 

tier  b  is 
—  2  k't- 
jinations 

arc  Ca 
h  (I,  otr. 
f  the  last 

tiers,  wo 
al  letters 


{a) 


>  find 


/     n    \ 
\s  -\-U' 

t  ni ember 
ators  are 
irs.     Mill- 
terms  \u 


—  l)n 


1 


VOMlilNATlONa, 


201 


I 


Tlu>  stiulrnt  is  now  rrcomnicndcd  to  pn  ovor  tlio  prorodinp  process 
will)  H|)cciul  siin|)U>  nuincricul  values  of  n  and  s  which  ht;  may  Hclect  for 
hiiuHiilf, 


3.4^5 

3   "' 


EXAMPLES. 

If  n  =  5  aiitl  H  =z  2,  wo  have 

1.3  4.  0.3  4.  3.4  _ 

U  n  =  7  and  .s-  =  3. 

4'')-C,.7 
1-2.3  +  2.3.4  +  3.4.5  -H  4. 5-0  =  _-— 1. 

4 

If  M  =  7  and  s  =  4, 

1.2-3.4  4-  2.3.4.5  +  3.4.5.0  =  ^—f'-^. 

o 


If  w  =  9  and  s  =  3, 
1.2.3  +  2.3.4  +  3.4.5  +  4.5.0  +  5.0.74-0.7.8  = 


0.7-8.9 


Prove  these  equations  by  com|Hiting  both  members. 

2G1.  Another  curious  example  is  the  following: 

Let  us  haVe  p  +  q  things  divided  into  two  sets,  the  one 
containing  />  and  the  other  q  things.  Tlicn,  to  form  all  possi- 
ble combinations  of  s  tbings  out  of  the  whole  />  +  q,  we  may 
take  : 

Any  s  things  in  set  /) ; 

Or  any  combination  of  s  —  1  things  in  sety;  with  any  one 
thing  of  set  q ; 

Or  any  combination  of  s  —  2  things  in  set^j  with  any  com- 
bination of  2  things  in  // ; 

Or  any  combination  of  s  —  3  things  in  ])  with  any  3  out 
of  q,  etc. 

We  shall  at  length  come  to  the  combinations  of  all  s  things 
out  of  q  alone.  Adding  up  these  separate  classes,  we  shall 
have : 

c?  +  cl^  a  +  CI2  cli  +  ....  +  cl  cl  1  +  cl 

This  sum  makes  up  all  combinations  of  s  things  in  the 
whole  p-\-q,  and  is  therefore  equal  to  6'«'^.  Putting  the 
numerical  expressions  for  the  combinations,  we  have  the 
theorem : 


m 


m 


Hi 


') 


2U2 


WMUINATI0N8. 


If,  fts  ail  oxamplo,  wo  put  s  =  3,  ^j  =  4,  «/  =  5,  this  tlieo- 


rem  will  give 
9.8.7 

i.a-3 


4-3.2       4.3  T)       4  r».4       r).4-3 


1.2.3  ■•"  1 


.2'l  "^  i'i.2  ■*■  1-2.: 


r 


tlio  curroctuuss  of  which  is  cusily  proved  by  computatioii. 

EXERCISES. 

1.  Write  all  the  coni])inations  of  tlirce  lefters  out  of  the 
five,  r/,  h,  c,  d,  e,  and  show  that  C^  of  tiiein  begin  wilh  n,  ('^ 
with  b,  and  C'j  with  c,  according  to  the  reasoning  of  §  2(jO. 

2.  Prove  that  C'»  =  C^  +  Cj, 

^4    —    ''4   ^    ^3> 

and  in  general,  6'"'^  =  C"  -\-  C'l-i. 

In  the  fv>llowir'T^  two  ways: 

(1.)  Let  all  combinations  of  s  letters  in  the  w  letters 

a,  b,  c, ....  71 

be  formed,  their  number  being  C'^.  Then  suppose  one  letter 
added,  making  the  nundjer  11  +  1.  The  combinations  of  s 
letters  out  of  these  n  -\-  1  will  include  the  C^  formed  from 
the  71  letters,  ])lus  each  combination  of  the  additional  (w  +  l)*' 
letter  with  the  combinations  of  s  —  1  out  of  the  first  71  letters. 

(2.)  Prove  the  same  general  result  from  the  formula, 


C 


'  =  (")• 


3.  If  we  form  all  combinations  of  3  things  out  of  7,  how 
many  of  these  combinations  will  contain  a  7,  and  how  many 
will  not? 

4.  If  we  form  all  the  combinations  of  s  letters  out  of  the  tz 
letters 

Wj     Of     Cy     •     •     •     •     ttj 


.. 


I 


coMiUNA  rroNs. 


293 


[lis  the'o- 


lOII. 


nt  of  tho 
itU  n,  ('I 


crs 


one  letter 
tions  of  8 
med  from 
il  {n  +  iy^ 
t  n  letters. 


lula, 


of  7,  how 
how  many 

ut  of  the  n 


how  many  of  these  combinations  will  contain  a,  and  how  many 
will  not  ? 

5.  In  tho  preredin;^  case,  how  many  of  the  comhinations 
will  contain  tho  three  letters  a,  b,  c? 

20'i.  TirmuKM  I.  TJir  total  tninihcr  of  mnilfuintlonft 
irhirli  van  Ite  fovDicd  from  n  tlUn^s,  lududing  1  zero 
coinhiiKitiotiy  in  2". 

In  the  language  of  Algchra, 

t'o  -\r  C\  -{■  C'i  -{•...,  ■\-  Cn-\  -f  Ct  =  2  . 

Prnnf.  Let  us  ])e<rin  with  ']  things,  a,  h,  c,  and  let  us  call 
the  formal  zero  combination,  1  =  C'o.     Then  we  have 

Numhcr  =  1 
**  =3 
«        =  3 

Sum  =  8  =  2". 

Now  introduce  a  fourth  letter  d.  Tho  rombinationH  out  of 
the  four  things,  a,  h,  c,  d,  will  consist  of  the  above  S,  plus  tho 
8  atlditional  ones  formed  bv  writing  d  after  each  of  the  above 
eight.     Their  nu!nber  will  therefore  be  10. 

In  the  same  way,  it  may  be  shown  that  we  double  tho  pos- 
sible number  of  combinations  for  every  thing  Ave  add.  to  the 
set  from  which  they  are  taken.    We  have  found,  for 

n  =  3,  Sum  of  combinations  =      8  =  2^; 

71  =  4,  «                 «              =  2-8  =  24; 

w  =  5,  «                 "              =  2-2*=  2'''; 

etc.  etc. 

which  shows  the  theorem  to  be  general. 

Theorem  II.  If  the  signs  of  the  aJtrrnate  combina- 
tions of  n  things  he  changed,  the  algebraic  sum  will  be 
zero. 


l)lank, 

cl 

a,  h,  c. 

cl 

ab,  ac,  br, 

ci, 

abc, 

In  algebraic  language, 


Hn 


Co  -  C'l  +  C'i  -C^  +  etc.  ±  Cl  =  0. 


(a) 


I 


294 


COMBINATIONS. 


Proof.    If  in  the  formula  of  §  201,  Ex.  2,  namely, 

/-^«+l  fin    .     p,n 

wc  put  n  —  1  for  n,  it  becomes 


Putting  8  successively  equal  to  0,  1,  2,  ...  .  n,  we  have 

rf«  pi^  1  . 


C^=  6' 


n-l 
1 


-.n-1 . 


+  cr' 


»i 


^  n-1  —    t  «-2  +  ^  n-l  —    f-  n-2  +  J-. 

Suhstitutiii,'  these  values  in  the  expression  (a),  it  becomes 

1  _  (i  +  cV)  +  (cr'  +  c^r')  -  (cr'  +  cv)  + . . . . 


=  1  _  1  _  cr' 


+  cr'  +  c 


6'r'  -  CV  +  etc. 


How  far  soever  wc  carry  this  process,  all  the  terms  cancel 
each  other  except  the  last.  Therefore,  if  we  continue  the  addi- 
tions and  subtractions  until  we  come  to  Cn-i ,  the  sum  will  be 

CIS  -C'l^C'l-  etc ±  Cl-i  =  ±  CT\  =  ±1. 

The  last  term  will  be  T  €„  =  T  1,  and  will  therefore 
just  cancel  the  sum  of  the  preceding  terms. 

Note.  Theorem  I  may  be  demonstrated  by  these  same  formulae, 
pincc  the  sum  of  all  the  terms  taken  positively  will  be  dui)licated  every 
time  we  increase  7i  by  1. 

263.  Independent  Combinnfions.  There  is  a  system  of 
combinations  formed  in  the  following  way  : 

It  is  reqidrcd  to  form  a  comhinatioih  of  s  tJiiji^s,  hy 
taking  one  out  of  each  of  s  different  collections.  How 
many  combinations  can  he  formed  ? 

Let  the  1st  collection  contain  a  things, 


i( 

2d 

a 

a 

h 

(I 

<( 

3d 
etc. 

n 

(i 

C 

etc. 

<< 

'i 


\m 


COMELY A  TIONS. 


295 


we  have 


t  becomes 


-yll—l 
^3 


-|-      .     •     •     • 

+  etc. 

rms  cancel 
e  the  ad  di- 
ll m  will  be 

=  ±1. 

Il  therefore 

|no  formula?, 
licated  every 

system  of 
\fiis.    lluw 


i 


Then  wo  may  take  any  one  of  a  things  from  the  first  col- 
lection. 

"With  each  of  these  we  may  combine  any  one  of  the  ?^  tilings 
in  the  second  collection. 

With  each  of  these  we  may  combine  any  one  of  the  c  things 
of  the  third  collection. 

Continuing  the  reasoning,  we  see  that  the  total  number  of 
combinations  is  the  continued  product 

(the  ....  to  s  factors. 

If  the  number  in  each  collection  is  equal,  and  we  call  it  a, 
the  numl)cr  (»f  combinations  will  be  «*. 

This  form  of  combinations  is  that  which  corresponds  most 
nearly  to  the  events  of  life,  and  is  applicable  to  many  questions 
concerning  probabilities.  For  example,  if  any  one  of  five  dif- 
ferent events  might  occur  to  a  ])crson  every  day,  the  numl)cr 
of  different  ways  in  which  his  history  during  n,  year  might  turn 
out  is  5'"^,  a  number  so  enormous  that  255  digits  would  be  re- 
quired to  express  it. 

EXERCISES. 

1.  A  man  driving  a  span  of  horses  can  choose  one  from  a 
stud  of  10  horses,  and  the  other  from  a  stud,  of  12.  llow 
many  different  span  can  he  form  ? 

2.  It  is  said  that  in  a  general  examination  of  the  public 
schools  of  a  county,  the  pupils  spelt  the  word  scholar  in  230 
different  ways.     If  in  spelling  they  might  replace 

ch  by  c  or  ^* ; 

0  by  an,  aic,  or  oo\ 

1  by  //; 

a  by  e,  o,  ii,  or  on ; 
r  by  re ; 
in  how  many  different  ways  might  the  word  be  spelt? 

3.  If  a  coin  is  thrown  n  times  in  succession,  in  how  many 
different  ways  may  the  throws  turn  out  ? 

4.  If  there  are  three  routes  between  each  successive  two  of 
the  five  cities,  Boston,  New  York,  Philadelphia,  Baltimore, 
Washington,  by  how  many  routes  could  we  travel  from.  Boston 
to  Washington? 


I, 


206 


COMBINA  TIONS. 


'I 


;»' 


The    »5iiioiniiil  Tliooroin  whou  the  Power  is  a 

AVliole  Number. 

204.  The  binomial  theorem  (§  173),  when  tlie  power  is  a 
positive  integer,  cim  be  demonstrated  by  the  doetrine  of  com- 
binations, as  follows: 

Let  it  first  he  required  to  form  the  product  of  the  n 
hinuniial  factors, 

To  understand  the  form  of  the  product,  let  us  first  study  the  special 
case  when  «-  =  o.  Performing  the  multiplication  of  the  first  three  fac- 
tors, the  product  will  consist  of  eight  terms : 


rtl^2^3    +   ^1^2-'''3    +   ^<l«3-'^'3    +  (^Z^Ti^t    +   "  \^' 2^' Z 

This  product  is  the  expression  («)  developed  when  n  —  3. 


{a) 


Wc  conclude,  by  induction,  that  the  entire  product  {a) 
when  develoi)cd  in  this  s^anie  way,  will  be  composed  of  a  sum 
of  terms,  each  term  l)eing  a  product  of  several  literal  factors. 

When  (a)  is  thus  multiplied  out,  we  shall  call  the  result 
the  developed  express  ion. 

The  developed  expression  has  the  following  properties  : 

I.  Each  term  contains  n  literal  factoids,  a's  and  x's, 
and  no  more. 

For,  suppose  aj^  =  rrj,  3*3  =^3,  to  Xn  =  an'  Then  the 
expression  {a)  will  reduce  to     ' 

'"Z^a^a.^n^  .  ...  an,  (b) 

and  the  developed  expression  must  assume  the  same  value ; 
that  is,  it  must  consist  of  terms  each  of  which  reduces  to  the 
expression 


(fi(Uf>:\ 


.  (f 


»» 


(c) 


when  we  change  x  into  a.  Now  if  it  contained  any  term  with 
either  more  or  less  than  it  factors,  it  could  not  assume  this 
form. 


> 


/. 


T  is  a 


)wcr  IS  a 
of  com- 


of  the  n 


the  special 
t  three  fac- 


(«') 


rod  net  {a) 
I  of  a  simi 

factors. 

the  result 

ertics  : 
and  -^'s, 

Then  the 

ame  value  ; 
noes  to  the 

(«) 

■j  term  with 
assume  this 


THE  BINOMIAL    THEOREM. 


297 


II.  TJie  factoids  of  each  tei'ni  have  all  the  n  indices 

Xf      /ilf     Of    •    •    •    •    71, 

For,  the  index  figure  of  no  term  is  altered  by  changing  x 
into  a,  as  in  I.  Hence,  if  in  any  term  any  index  figure  were 
missing  or  repeated,  that  term  would  not  reduce  to  the  form 
(('),  whence  tliere  can  be  neitiier  omission  nor  repetition  of 
any  index. 

III.  Because  each  term  has  n  factors^  it  Tiiust  either 
have 

n  factors  a; 

n  —  1  factors  a  and  one  factor  x; 
n  —  3  factors  a  and  two  factors  x; 
In  general,  a  term  may  hrive  the  factor  a  repeated 
n  —  i  times,  and  x  rrj)eated  i  times. 

IV.  In  a  term  which  contains  i  factors  x,  these  i  Victors 
must  be  affected  with  some  combination  of  i  indices  out  of  the 
whole  number  1,  3,  3,  ....  w  ;  and  the  n  —  i  n's  must  bo 
alTected  by  the  complementary  combination  of  n  —  i  indices. 
We  next  inquire  whether  tliere  is  a  term  corresponding  to 
every  such  combination.     Let 

Aj      Oj     *xj      1 9    •    •    •   • 

be  any  combination  of  i  indices,  and 

/w,     O,    Uj    O,  .  .  .   . 

the  complementary  combination  of  7i  —  i  indices. 

Since  the  developed  expression  must  be  true  for  all  values 
of  rt  and  X,  let  us  put  in  {a), 


«i 


0, 


«3  =  ^^ 


«4 


0, 
0, 


etc. 


X..  =  0; 
ajg  =  0; 
x^  =  0; 
X,  =0; 
etc. 


(^0 


The  product  {a)  will  then  reduce  to  the  single  term, 

XiffoX^x^n^af^x^a^ (e) 

By  the  same  change  tlui  developed  expression  must  reduce 
to  this  same  value,  and  it  cannot  do  this  unless  the  expression 
(e)  is  one  of  its  terms. 


■a\} 


H 


298 


COMBINATIONS. 


Hence  the  fleveloped  exprcssinn  must  contain  a  ternv 
coTvcspoiidiu^  to  every  eonihitintion. 

V.  Since  every  combination  of  /  figures  out  of  1,  2,  3, ....  7i 
will,  in  this  way,  give  ri^^e  to  a  term  like  {p).  containing  the 
symbol  a  i  times,  and  the  symbol  x  n  —  i  times,  there  will  be 
6'?  such  terms. 

Now  suppose      rTj  =  a^  =  r/3  =  .  . 


\ 


(In  =  a. 


■'\    =   ^'2    =    '>'3    = 


The  expression  (a)  will  then  reduce  to  {a  -f-  x)"^. 

In  the  developed  expression,  all  the  C'i  terms  containing  x 
I  times  and  a  n  —  i  times  Avill  now  be  equal  and  their  sum 
Avill  reduce  to  C"  a"'    x . 

Hence,  putting  in  succession  i  =  0,  i  =.  1,  etc.,  to  i  =  ??, 
we  shall  have 

{a  4-  >•)"  =  (i""  4-  Ci  r/'^-i  x  4-  Cl ««"  2  a:2  -f +  C' Ji  1  a./"-i  +  x\ 

Substituting  for  C'f  its  value,  we  shall  have 

{a  +  xY  =  «"  +  na^'-^x  -\-  l^i'^-'^x-  +  ....  +  (^^  Xix"-'^  +  yjx\ 

which  is  the  Buio??iial  77/ coirm,  enunciated,  but  not  demon- 
strated, in  Book  V,  Chapter  I. 

Note.  If  the  stiulont  has  any  diffioulty  in  nnrlprstandinp  the  steps 
of  the  precedintr  demonstration,  he  should  suppose  n  =  3,  and  refer  the 
demonstration  to  the  developed  expression  («'). 


r 


•I 


PROBABILITIES. 


299 


a  term 


3, n 

iiinp:  the 
e  will  be 


} 


taining  x 
heir  sum 

to  i  —  n, 
t  demon- 


p:  the  steps 
d  refer  the 


CHAPTER    III. 

THEORY    OF     PROBABILITIES. 

205.  Def.    The  Theory  of  Probabilities  treats  of 
the  chanc*?s  of  the  occurrence  of  events  which  cannoc  be  * 
foreseen  with  certainty. 

Nofation.  Let  a  hix^  contciin  4  balls,  of  which  1  is  white 
and  3  black.  If  a  ball  be  drawn  at  random  from  the  bag,  we 
should,  in  ordinary  language,  say  that  the  chances  were  1  to  3 
in  favor  of  the  ball  being  white,  or  3  to  1  in  favor  of  its  being 
black. 

In  the  language  of  iDrobabilities  we  say  that  the  probability 

1  3 

of  a  white  ball  is  -r,  and  that  of  a  black  one  -• 
4  4 

In  general,  if  there  are  m  chances  in  favor  of  an  event,  and 


n  chances  against  it,  its  probability  is 


m 


Hence, 


in  +  n 

Def.  The  Probability  of  an  event  is  the  ratio  of 
the  chances  whi^h  favor  it  to  the  whole  number  of 
chances  for  and  against  it. 

lUusf rations.     If  an  event  is  certain,  its  probability  is  1. 
If  the  chances  for  and  against  an  event  are  even,  its  prob- 
1 


ability  is 


2 


If  an  event  is  impossible,  its  probability  is  0. 

Cor.  1.     If  the  probability  that  an  event  will  occur 
is^,  the  probability  that  it  will  fail  is  1  —p. 

Car.  2.  A  probability  is  always  a  i)ositive  fraction, 
greater  than  0  and  less  than  1. 

266.  Method  of  Prohahiliiies.     To  find  the  probability  of 
an  event,  we  must  be  able  to  do  two  things: 


300 


PROBABILITIES. 


i  I 


M 


1.  Enumerate  all  possihle  irnj/s  in  irhicli,  the  event 
mnij  oeciir  or  fail,  it  heiit£  snf)/)oscd  that  these  ways 
are  all  eqaally  probable. 

2.  Deternilne  hoiu  viany  of  these  ways  irill  lead  to 
the  event. 

If  n  be  the  total  number  of  ways,  and  m  the  number  whicli 

7)1 

lead  to  the  event,  the  probability  required  is  —  • 

EXERCISES. 

1.  A  die  has  2  white  and  4  black  sides.  What  is  the  prob- 
ability of  throwiuf^  a  white  side  ? 

2.  A  bag  contains  u  balls  numbered  from  1  to  w,  the  even 
numbers  beiug  white  and  the  odd  ones  black.  What  is  the 
probability  of  drawing  a  black  ball  when  n  is  an  odd  number? 
AVhat,  when  ?i  is  an  even  number  ? 

3.  A  bag  contains  3«+2  balls,  of  which  numbers  1,4,  7, 
etc.,  are  white  ;  2,  5,  8,  etc.,  arc  red;  3,  G,  9,  etc.,  are  black. 
What  are  the  respective  probabilities  of  drawing  a  Avliite,  red, 
and  black  ball  ? 

Rem.    In  the  last  example  the  probabilities  are  all  less  than     ;  there- 
in 

fore,  should  one  attempt  to  guess  the  color  of  the  ball  to  be  drawn,  ho 
would  be  more  likely  to  be  wrong  than  right,  no  matter  what  color  ho 
guessed.  This  exemplifies  a  lesson  in  i)ractical  judgment  to  be  drawn 
from  the  theory  of  probabilities.  If  there  are  three  or  more  possible  re- 
sults of  any  cause,  it  may  happen  that  the  best  judgirient  would  be  more 
likely  to  be  wrong  than  right  in  attempting  to  jiredict  the  result.  Thus, 
if  there  are  three  presidential  candidates  with  nearly  eijiial  chances,  the 
chances  would  be  against  the  election  of  any  one  that  might  be  nanu^d. 

Gamblers  of  the  turf  are  nearly  always  found  betting  odds  against 
every  hnrse  that  may  be  entered  for  a  race,  though  it  is  certain  that  one 
of  them  will  win. 

Hence,  if  a  natural  event  may  arise  from  a  number  of  causes  with 
nearly  equal  facility,  it  is  unphilosophical  to  have  any  theory  whatever 
of  the  cause,  b(>cause  the  chances  may  be  against  the  most  probable 
cause  being  the  true  one. 

Probabilities  tlepeiidiiig'  upon  Conibiiiatioiis. 

307.  Probirm  i.  Two  coins  arc  thrown.  What  are  the 
respective  probabilities  that  the  result  will  be  :  Both  heads? 
head  and  tail  ?  both  tails  ? 


t 


1) 


PROBABILITIES. 


301 


rj  eve  tit 
e  ways 

lead  to 

er  wh  ick 


he  prob- 

the  even 
at  is  tlio 
number? 

s  1,  4,  7, 
re  black, 
bite,  red, 

\     ;  there- 
in 
rawn,  ho 

color  ho 

be  drawn 

)Of*Riblo  re- 

hc  more 

.     Thus, 

lancos,  the 

named. 

ds  ngiiinst 

that  one 

auses  with 
whatever 
probable 


tions. 

t  are  the 
b  heads  ? 


lit 


i 


At  Ib'st  sight  it  might  ajipcar  Hiat  tbo  chanees  in  favor  of 
these  three  results  were  equal,  and  that  therefore  the  probabil- 
ity of  each  was  ^'     But  this  would  be  a  mistake.     To  find  the 

probabilities,  wo  must  combine  the  possible  tbrows  of  the  first 
coin  (\vlii(;h  call  A)  with  the  possible  throws  of  tbc  second 
(which  call  13),  thus  : 

A,  head  ;  B,  head. 

A,  head  ;  B,  tail. 

A,  tail  ;  B,  head. 

A,  tail ;  B,  tail. 

These  combinations  arc  all  equally  probable,  and  while 
there  are  only  one  each  for  both  heads  and  l)oth  (ails,  there  are 

two  for  head  and  tail.     Hence  the  probabilities  are  -,  -,    ,• 

■i        /v         "i 

The  sum  of  these  three  probabilities  is  1,  as  it  ouglit  always 
to  be  when  all  possible  results  are  considered. 

rroh.  2.  Five  coins  are  thrown.  What  are  the  resiiectivo 
probabilities:         q  heads,  5  tails? 

1  head,  4  tails? 

3  heads,  3  tails? 

etc.  etc. 

Let  the  several  coins  be  marked  a,  b,  c,  lU  c.  Coin  a  may 
be  either  head  or  tai',  making  two  cases.  Each  of  these  two 
cases  of  coin  a  may  he  combined  with  either  case  of  d  (as  in  the 
last  exami)le),  making  4  cases. 

Each  of  these  4  cases  may  be  combined  with  either  case  of 
coin  c,  making  8  cases. 

Continuing  the  process,  the  total  number  of  cases  for  five 
coins  is  2'  =  33. 

Of  these  32  cases,  only  one  gives  no  head  and  5  tails. 

There  are  5  cases  of  I  head,  namely:  a  alone  head,  b  alone 
head,  etc.,  to  r. 

2  heads  may  be  thrown  by  coins  <?,  b ;  a,  c,  etc. ;  b,  c;  b,  d, 
etc. ;  c,  d,  etc. ;  that  is,  by  any  combination  of  two  letters  out 
of  the  five,  a,  b,  c,  d,  e.     llence  the  number  of  cases  is 

Cl  =  10. 


1   m 

iii 


303 


PROBABILiriES. 


m 


'\  v^% 


In  the  same  way  tlio  immbor  of  cases  corresponding  to  3, 
4,  and  5  heads  are,  respectively, 

Cl  =  10,         C\  =  5,         Ci  =  1. 

Dividinfi:  by  the  whole  nnmber  of  cases,  we  find  the  respec- 
tive probabilities  to  be 

1  .  5       10,         10       5        1 

35'        32'     32'        32'     32'    32* 

The  followin,2^  general  proposition  is  now  to  be  proved  by 
the  student : 

Theouem.  //  thrj'e  are  n  coins,  the  probabilitjj  of 
throiv'ui^  s  heads  and  n  —  s  tails  is 

2«" 

From  this  resnlt  we  may  prove  the  theorem  in  combina- 
tions of  §  262.  If  we  suppose,  in  succession,  *•  =r  0,  s  =  1, 
s  =  2,  etc.,  to  5  =  71,  the  respective  probabilities  of  0  head, 
1  head,  2  heads,  etc.,  will  bo 


Cl      0}      Cl 
g/i '     2'^ '     2"  ' 


etc.,    to 


CJ 

2«* 


Because  the  sum  of  all  these  probabilities  must  be  unit}", 
we  find 

C?  +  C?  4-  C^  +  ....  +  C;:  =  2^ 

Proh.  3.  Two  dice  are  thrown  at  backgammon.  What  arc 
the  respective  probabilities  of  throwing  5  and  0  and  two  G's  ? 

If  we  call  the  dice  a  and  h,  any  number  from  1  to  0  on  rt 
may  be  combined  with  any  number  from  1  to  6  on  h.  There- 
fore, there  are  in  all  36  possible  combinations. 

In  order  to  throw  two  G's,  a  must  come  6  and  h  also. 
Therefore  there  is  only  one  case  for  this  result,  so   that  its 

probability  is  --• 

To  bring  5  and  6,  a  may  be  .5  and  h  0,  or  l  5  and  a  6.  So 
there  are  two  cases  leading  to  this  result,  and  its  probability  is 

36  ~  18* 


i 


is  (  •  i 


rnOB  ABILITIES. 


303 


I  to  3, 


rcspcc- 


ved  by 
ity  of 


imbina- 
0  head. 


c  unitv, 


hint  arc 

|()  O's? 
G  on  a 
Thcre- 

h  also. 
I  that  its 


G.     So 
Ibility  is 


Note.  That  5  and  0  arc  twico  as  probable  ns  a  double  G  may  bo 
clearly  seen  by  supposiiif?  that  the  two  dice  are  thrown  in  succe.sHion.  If 
the  lirst  throw  is  either  5  or  (5,  there  is  a  chance  for  the  combination  5,  6, 
but  there  is  no  cliance  for  a  double  0  unless  tlie  lirst  throw  is  G. 

Proh.  4.  If  three  dice  are  tlirown,  what  are  tlie  respective 
probabilities  that  the  numbers  will  be: 

1,  1,  1?  1,  1,  3?  1,  2,  3? 

The  solution  of  this  case  is  left  as  an  exercise  for  the 
student. 

Prob.  5.  From  a  hiv^  containing  3  white  and  2  l)lark  balls, 
2  balls  are  drawn.     What  are  the  respective  probabilities  of 

Botli  balls  white? 
1  white  and  1  black  ? 
Both  black  ? 

Since  any  2  balls  out  of  5  may  be  drawn,  the  total  number 
of  eases  is  6*2- 

Only  one  of  these  combinations  consists  of  two  white  balls. 

C'a  of  the  cases  bring  both  ])alls  black. 

A  white  and  black  are  formed  by  coml)ining  any  one  of  the 
three  white  with  any  one  of  the  two  black. 

The  respective  probal)ilities  can  now  be  deduced  by  the 
student. 

EXERCISES. 

1.  It  takes  two  keys  to  unlock  a  safe.  They  are  on  a 
bunch  with  two  others.  The  clerk  takes  three  keys  at  random 
from  the  bunch.  What  is  the  probability  that  he  has  both  the 
safe  keys? 

2.  A  party  of  three  persons,  of  whom  two  are  brothers,  seat 
themselves  at  random  on  a  bench.  What  are  the  probabilities 
{a)  that  the  brothers  will  sit  together,  {h)  that  they  will  have 
the  third  man  between  them  ? 

3.  If  two  dice  are  throw^n  at  ])ackgammon,  Avhat  are  the 

probabilities 

{a)  Of  two  aces  ? 

(5)  Of  one  ace  and  no  more  ? 

4.  In  order  that  a  player  at  backgammon  may  strike  a  cer- 


il 


■! 


304 


Vli(HiMilUTlIiJ8. 


% 


\  * 


'I 


If  lif 


tain  poini,  the  sum  of  tlio  mnnbers  thrown  must  bo  8.     What 
are  his  chances  of  siicccetling  in  one  throw  of  his  two  dice  ? 

5.  A  party  of  1:5  persons  sit  at  a  round  table.  AVhat  is  the 
prohubiUly  tluit  .Mr. 'I'aylor  and  Mr.  Williams  will  be  next  to 
each  other  ?     (See  §  •Zb'i.) 

6.  An  illiterate  servant  puts  two  works  of  3  volumos  each 
npon  a  shelf  at  random.  What  is  the  probability  that  both 
pair  of  companion  volumes  are  together? 

7.  A  gentleman  having  three  pair  of  boots  in  a  closet,  sent 
a  blind  valet  to  bring  him  a  pair.  The  valet  took  two  boots  at 
random.  What  are  the  chances  that  one  was  right  and  the 
other  left  ?     What  is  the  probability  that  they  were  one  pair? 

8.  If  tho  volumes  of  a  J3-volume  book  are  jdaced  at  random 
on  a  shelf,  what  is  the  })robability  that  they  will  be  in  regular 
order  in  either  direction  ? 

9.  A  man  wants  a  particular  span  of  horses  from  a  stud 
of  8.  His  groom  brings  him  5  horses  taken  at  ran{U)m.  AVhat 
is  the  probability  that  both  horses  of  the  span  are  amongst 
them  ? 

10.  From  a  box  containing  5  tickets,  nuvdiercd  1  to  5, 
3  are  drawn  at  random.  Wiiat  is  the  probability  that  numbers 
2  and  b  arc  both  amongst  them  ? 

11.  The  same  thing  being  supposed,  Avhat  is  the  probability 
that  the  sum  of  the  two  numbers  remaining  in  the  box  is  G  ? 

12.  Of  two  purses,  one  contains  5  eagles  and  another  10 
dollar-pieces.  If  one  of  tho  purses  is  selected  at  random,  and 
a  coin  taken  from  it,  what  is  the  probability  that  it  is  an 
eagle  ? 

13.  From  a  bag  containing  3  white  and  4  black  balls 
2  balls  arc  drawn.  What  is  the  probability  that  they  are  of 
the  same  color  ? 

14.  The  better  of  two  chess  phiyers  is  twice  as  likely  to  win 
as  to  be  l.)eaten  in  any  one  game.  What  chance  has  his  weaker 
opponent  of  Avinning  2  games  in  a  match  of  3  ? 

15.  Fi'om  a  bag  containing  m  white  and  u  black  balls,  two 
balls  are  drawn  at  random.  What  is  the  probability  that  one 
is  white  and  the  other  black  ? 


rilOliAlilLlTlKH. 


305 


What 
ice  ? 
it  is  tho 
next  to 

lies  cat'li 
lut  both 

set,  sent 

boots  at 

and  tlio 
no  pairV 

b  random 
n  regular 

jm  a  stud 
n.  AVhat 
}  amongst 

il  1  to  5, 
It  numbers 

irobabiliiy 

»X  IS  (»  . 

nothcr  10 
idom,  and 
it  is  an 

iack   balls 
[icy  are  of 

telv  towin 

r    " 

[lis  weaker 

balls,  two 
that  one 


i6.  From  a  bag  containing  1  white,  3  red,  and  .'J  blaek 
balls,  3  balls  are  drawn.  What  is  the  |)robubility  that  they  uro 
all  of  ditlerent  colors  ? 

17.  It*  H  coins  are  thrown,  what  s  the  chance  that  thero 
will  he  one  head  and  no  more  ':* 

18.  From  a  Congressional  coniniitteo  of  0  Kepublicans  and 
5  Democrats,  a  sub-committee  ol"  '.I  is  chosen  by  lot.  What  is 
the  probal>ility  that  it  will  be  composed  of  two  Kepublicaud 
and  one  Democrat  'i 

Compound  Events. 

2(58.  Theorem  I.  The  ])rol)tihiH tij  that  tirn  iiulcpcnd' 
cub  events  will  both  h((])f)cii  is  C(/a((l  to  the  pruducb  of 
their  separate  prohabillties. 

Proof.  For  the  first  event  let  there  bo  m  cases,  of  which 
p  are  favorable;  and  for  the  second  h  cases,  of  which  y  are 
favorable.     Then,  by  definition,  the   respective   probabilities 

AvlU  be  —  and   -• 
m  n 

When  both  events  are  tried,  any  one  of  the  m  cases  may  bo 
combined  with  any  one  of  the  ii  cases,  making  in  all  ni  x  n 
combinations  of  equal  probaI)ility. 

The  combinations  favorable  to  both  events  Avill  be  those 
only  in  which  one  of  the  2^  cases  favorable  to  the  first  is  com- 
bined with  one  of  the  q  cases  favorable  to  the  second.  Tho 
num])er  of  these  coml)inations  is  p  x  y- 

Therefore  the  probability  that  both  events  will  happen  is 

m  X  n        m       n* 
which  is  the  product  of  the  individual  probabilities. 

If  there  are  three  events  of  which  the  probabilities  are  p,  q, 
and  r,  and  we  Avish  to  find  the  probability  that  all  three  will 
happen,  we  may  by  what  precedes  regard  the  concurring  of  the 
first  two  events  as  a  single  event,  of  which  the  probability  is 
pq.  Then  the  probability  that  the  third  event  will  also  con- 
cur is  the  product  of  this  probability  into  r,  or 

pqr. 
20 


iill 


i  .., 


300 


PliOIiAlill.iriKH. 


»l 


m 


Proceeding  in  the  same  way  with  4,  5,  0, ... .  events,  we 
reach  the  general 

Tiii;oui:m  II.  Tlia  prohdhililn  Ihat  mnj  iinuihrr  of  hi- 
driwiulcut  cAriits  irill  nil  uccfir  is  i'(/ii(tl  to  the  contimved 
product  of  their  iiidividiud  prob((bUities. 

Ukm.  This  theorem  is  of  great  practiciil  use  as  a  guide  to 
our  expectations.  It  teaches  that  if  success  in  an  enterprise 
re(|uires  the  concurrence  of  a  great  nuniher  of  I'avorahle  cir- 
cumstances, the  ciiances  may  he  greatly  against  it,  although 
each  circumstance  is  more  hkely  than  not  to  occur. 

This  is  ilhistrated  by  tiie  following 

Examplp:  I.  A  traveller  on  a  journey  by  rail  has  8  connec- 
tions to  make,  in  order  that  he  may  go  through  on  time. 
There  are  two  chances  to  one  in  favor  of  each  connection. 

What  is  the  probability  of  his  keeping  on  time  ? 

2 
The  probability  of  each  connection  being     ,  the  probabil- 

ity  of  successfully  making  the  tirst  two  connections  will,  by  the 

preceding  theorems,  be  ("I ,  the  first  three  P) ,  and  all  eight 

Therefore  tliere  arc  25  chances  to  1  a^'ainst  his  going 
through  on  time. 

On  the  other  hand,  if,  instead  of  any  one  accident  being 
fatal  to  success,  success  can  be  prevented  only  by  the  concur- 
rence of  a  series  of  accidents,  the  probability  of  failure  may 
become  very  small. 

Ex.  2.  A  ship  starts  on  a  voyage.  It  is  an  even  chance 
that  she  will  encounter  a  heavy  gale.    The  probability  that ' 

she  will  not  spring  a  leak  in  the  gale  is  -,-•     If  a  leak  occurs, 

9 
there  is  a  probability  of  :j^  that  the  engine  will  be  able  to 

pump  her  out.     If  they  fail,  the  probability  is  '.  iha-t  the  com- 


ivcnts,  we 

i/v  of  ift'- 
Diitliiivcd 

()nil)U'-  t'ir- 
;,  althougli 


s  8  cnnncc- 
h  on  time, 
connection. 

;ic  pro1)ul)il- 
will,  by  the 
d  all  eight 


t  his  going 

idcnt  being 
the  ronciir- 
iaihu-e  may 

even  chance 
)ability  that* 

leak  occurs, 
I  be  able  to 
hat  the  com- 


PROUABILiTIES.  807 

piirtnicnfs  will  keep  (he  ship  afloat.  If  slio  sinks,  it  is  an  even 
cliaiur  that  any  one  piussengcr  will  l»e  saved  l»y  the  hoats 
What  is  the  })rol>ability  tiiat  uuy  Individ uul  pa.sbun<jei'  will  bo 
loat  at  sea  ? 

The  prol)ability  that  • 

(he  ship  will  meet  a  heavy  galo  is _ 

the  ship  Avill  spring  a  leak  in  the  galo  is ^• 

the  engines  cannot  pi  nip  her  out  is — 

tho  compartments  cannot  keep  her  afloat  is - 

the  boats  cannot  save  the  passenger  is     .......  ^ 

The  continued  product  of   these   probabilities   is    ,  ,,-, 

1000 

which  is  the  probability  that  the  passenger  will  be  lost. 

2Gt).  The  preceding  theorem  as  enunciated  supposes  that 
ihc  several  events  are  indcpendoU,  that  is,  that  the  prohaljility 
of  the  occurrence  of  any  one  is  not  aflected  by  the  occurrence 
or  non-occurrence  of  the  others.  To  investigate  what  modifi- 
cation is  required  when  the  occurrence  of  one  of  the  events 
alters  the  probability  of  another  of  the  events,  let  us  distinguish 
the  two  events  as  i\\e first  and  second.     We  then  reason  thus: 

Let  the  total  number  of  equally  possible  cases  bo  m,  and  let 
p  of  these  cases  favor  the  first  event.     Its  probability  will 

then  be  —  • 
m 

It  is  certain  that  the  events  cannot  both  happen  unless  the 

first  one  happens.     Hence  the  cases  which  favor  both  events 

can  be  found  only  among  the  p  cases  wdiich  favor  the  first. 

Let  q  of  these  ^j  cases  favor  the  second  event.     Then  the  prob- 

q 
ability  of  both  events  will  be  — • 
*'  111 

In  case  the  first  event  happens,  one  of  the  ^;  cases  which 


I 


308 


PROBABILITIES. 


favor  it  must  occur,  and  the  probability  of  the  second  event 


'I 


will  then  be 


P 


Then 


Probability  of  both  events  =  --  =  —  x  -•    Hence, 

•'  '))j  •111  n 


Ul 


m 


TiiEOREM.  TliG  prohdhility  that  ttco  events  irUl  both 
occur  is  equal  to  the  prohabUitij  of  the  first  event  inultl- 
plicd  by  tlie  probabilitfj  of  the  second,  in  case  the  Jirst 
occurs. 

By  continuing  the  reasoning  to  more  evenU,  we  reach  the 
general 

THEOKE^r.  Tlie  probnhility  that  a  number  of  events 
tcill  all  occur  is  er/udl  to  the  product 

I  X  Prob.  of  second  in  case  first  occurs. 
Prob.  of  first  i  x  Prob.  of  third  in  case  first  two  occur. 

(  X  Prob.  of  fourth  in  case  first  three  occur, 
etc.  etc.  etc. 

Example.  From  a  bag  containing  2  white  and  3  black 
lialls,  2  balls  are  drawn.  What  are  the  probabilities  (1)  that 
both  balls  are  white,  {2)  that  both  are  black? 

This  problem  has  alreadv  been  solved,  but  we  are  now  to 
see  how  the  answers  may  be  reached  by  the  last  theorem.  It 
is  evident  that  we  may  sui)])Ose  the  two  balls  drawn  out  one 
after  the  other,  and  the  probabilities  of  their  being  white  or 
black  will  be  the  same  as  if  both  were  drawn  together. 

I.  Both  balls  white.     The  probability  that  the  first  ball 
2 
drawn  is  white  is  ^'    If  it  really  proves  to  be  white,  there  will 

be  left  1  white  and  3  black  balls.     In  this  event,  the  probability 

that  the  second  also  will  be  white  is  - 

4 

Hence  the  probability  that  both  are  white  is 

2       1  _   1 
5  ^  4  ~  l6' 


M 


i 


PROBABILJTfES. 


309 


20iid  event 


[leuce, 

will  both 
eni  inidti- 
jc  the  Jirst 

re  reach  the 

r  of  events 

occurs. 
s\'0  occur, 
three  occur, 
etc. 

and  3  black 
ilies  (1)  that 

\'c  are  now  to 

theorem.    It 

raAvn  out  one 

ing  white  or 

ther. 

the  first  hall 

Ate,  there  will 

he  probability 


< 


I 


II.  Itnfh  halls  hlarlc.    Applying  the  same  reasoning,  wo 
find  I'or  the  probability  of  tliis  case, 


-   X  -  =      -  • 

5       3        10 


EXERCISES. 


1.  Two  men  embark  in  separate  commercial  enterprises. 
The  odds  in  favor  of  one  are  3  to  2 ;  in  favor  of  tlie  other,  'i 
to  1.  What  are  the  probabilities  (1)  that  both  will  succeed? 
(2)  that  both  will  fail? 

2.  The  probability  that  a  man  will  die  within  ten  years  is 

-,  and  that  his  wife  will  die  is  — •  What  are  the  respective 
probabilities  that  at  the  end  of  ten  years, 

(«)  Both  are  living? 

(/3)  Both  are  dead  ^ 

(y)  Husband  living,  but  wife  dead? 

{6)  Ilusband  dead,  but  wife  living? 

2 

3.  The  probability  that  a  certain  door  is  locked  is  -•    The 

o 

key  is  on  a  bunch  of  4.  A  man  takes  2  of  the  four  keys,  and 
goes  to  the  door.  What  are  the  chances  that  he  will  be  able  or 
unable  to  go  through  it  ? 

4.  Two  bags  contain  each  4  black  and  3  white  balls.  A 
person  draws  a  ball  at  random  from  the  first  bag,  and  if  it  be 
Avhite  he  puts  it  into  the  second  bag,  mixes  the  balls,  and  then 
draws  a  ball  at  random.  What  is  the  probability  of  drawing 
a  white  ball  from  each  of  the  bags  ? 

5.  If  a  Senate  consists  of  m  Democrats  and  n  Republicans, 
what  is  the  probability  that  a  committee  of  three  will  include 
2  Democrats  and  1  Republican? 

6.  A  bag  contains  2  white  balls  and  5  black  ones.  Six 
people,  A,  B,  C,  D,  E,  F,  are  allowed  to  go  to  the  bag  in  alpha- 
betical order  and  each  take  one  ball  out  and  keej)  it.  The 
first  one  who  draws  a  white  ball  is  to  receive  a  prize.  Wliat 
are  their  respective  chances  of  winning? 


i 


Note. 
all  7  baUs. 


A'a  chance  is  eadilv  calculated,  because  he  has  the  draw  from 


I    "I 

1  i 

I    i 


310 


PKOIIAUILITIES 


•I 


m  n 


In  order  that  B  may  win,  A  must  first  fail.  Therefore,  to  find  B's 
probability  we  find  (1)  the  probability  that  A  fails,  (3)  the  probability  that 
if  A  faila  then  IJ  will  win.  We  then  take  the  product  of  these  probabili- 
tiea. 

In  order  that  C  may  giiin  the  prize,  (1)  A  must  fail,  i2)  B  must  fail, 
(3)  C  himself  must  gain.  So  we  find  the  successive  probabilities  of  the.se 
occurrences. 

Continuing  to  F,  we  find  that  he  cannot  win  unlcps  the  5  men  before 
him  all  miss.  He  is  then  certain  to  gain,  because  only  the  two  white 
balls  would  be  left. 

7.  Two  nicii  hiive  one  throw  each  of  a  cohi.  X  olfei's  a 
prize  if  A  throws  head,  and  if  he  fails,  but  not  otherwise,  B 
may  try  for  tiie  prize.  If  botii  fail,  X  keeps  the  prize  himself. 
What  are  the  respective  chauccs  of  the  three  men  having  the 
prize  ? 

8.  A  and  B  arc  alternately  to  throw  a  coin  until  one  of 
them  throws  a  head  and  becomes  the  winner.  If  A  has  the 
first  throw,  what  are  their  respective  chances  of  winning  ii* 

9.  A  crowd  of  n  men  are  allowed  to  throw  in  the  same  way 
for  a  prize,  in  alpluibetical  order,  the  game  ceasing  as  soon  a.s  a 
head  is  thrown.  What  are  the  respective  chances  of  the  con- 
testants? 

10.  Three  men  take  turns  in  throwing  a  die,  and  he  who 
fir.<t  throws  a  G  wins.     What  are  their  respective  chances:' 

11.  If  4  cards  are  drawn  from  a  pack  of  52,  show  that  the 
probability  that  there  will  be  one  of  each  of  the  four  suits  is 

39  20  13 

5l'50'4ij* 

12.  One  purse  contains  5  dimes  and  1  dollar,  and  another 
contains  G  dinu'S.  5  i)ieces  are  taken  from  the  iirst  purse  and 
i)ut  into  the  second,  and  after  bein^f  mixed  5  are  taken  from 
the  second  and  put  into  the  first.  Whicli  purse  is  now  most 
likely  to'contain  the  dollar  ? 

13.  Of  two  purses,  one  contains  4  eagles  and  2  dollars,  the 
other  4  eagles  and  G  dollars.  One  being  taken  at  random,  and 
a  coin  drawn  from  it,  wl  .it  arc  the  respective  probabilities 
that  it  is  an  eagle  or  a  dollar? 


II 


PROBABILITIES. 


311 


to  find  B'3 
,ability  tliat 
e  prubiibili- 

lies  of  tlK'se 

,  men  before 
,ie  two  white 

X  offers  a 
thcrwisc,  B 
•ize  hini!^^'^^- 

having  the 

until  one  of 
If  A  has  tlio 
duniug  V 
the  same  way 
0-  as  soou  as  a 
s  of  the  cou- 

and  lie  who 
chances  V 

low  that  the 
bur  suits  is 


and  anotlier 
h-st  purse  and 
ve  taken  from 
}  is  now  most 


I 


2  dollars,  the 
t  random,,  and 
e  prohabiiilies 


Cases  of  *Uiieqiial  Probability. 

270.  Def.  If  two  or  more  possible  events  ai-e  so 
related  that  only  one  of  them  can  happen,  tliey  are 
called  Mutually  Exclusive  Events. 

Theouem.  Tlie  prohahUitii  tJutt  some  one  of  several 
exclusive  events,  we  care  not  which,  will  occur,  Is  equal 
to  the  sum  of  their  separate  probabilities. 

Proof.     Lot  there  be  m  possible  and  equally  probable  cases 

in  all;  let  p  of  these  cases  be  favorable  to  one  event,  q  to  tlic 

p  (I  r 

second,  r  to  the  third,  etc.,  so  that  — ,  — ,  — ,  are  the  ro- 

,.  ,    ,.,.,.  m  m  7ti' 

spective  probabihties. 

Since  only  one  of  the  events  is  possible,  the  p  cases  which 
favor  one  must  be  entirely  different  from  tiie  rj  cases  which 
favor  the  second,  and  these  cases  p-\-q  must  be  entirely  differ- 
ent from  the  ;•  which  favor  the  third,  etc. 

Hence  there  will  be  /;  +  r/  +  ;'  +  etc. ,  cases  which  favor  some 
one  or  another  of  the  events.  Hence  the  probability  that  some 
one  of  these  events  will  occur  is 

m  ' 

which  is  equal  to  the  sum  of  the  probabilities, 

par 

—  +  ~  -{ 1-  etc. 

m      m      m 

IiE>[.  If  the  concurrence  of  some  two  events,  say  the  first 
and  second,  had  been  possible,  some  one  or  more  of  the  j)  cases 
which  favor  the  first  would  have  been  found  amonj^  the  q  cases 
which  favor  the  second.  Then  the  whole  numl)er  of  cases 
Avhicli  favored  either  event  would  have  been  less  than  ])  +  q, 
and  the  probability  that  one  of  the  two  events  would  happen 
less  than  the  sum  of  their  respective  probabilities. 

271.  General  Problem.  To  find  the  j)robabilitj/  that 
an  event  of  whicli  the  probability  on  any  one  trial  Is  p, 
will  happen  exactly  s  times  in  n  trials. 


4i 


fl 


312 


PUOBABILITIES. 


'I 


This  prnhlem  is  at  (lie  basis  of  some  of  the  widest  ajiplica- 
tioiis  of  the  theory  of  prol)al)ility  to  pnictical  questions,  e.'^jjc- 
cially  tliose  assoeiated  with  life  and  tire  insurance.  The  con- 
ditions which  it  imjilies  are  therefore  to  be  fully  comprehended. 

We  may  conceive  a  trial  to  mean  fjicimj  the  event  an  oppor- 
tunitrj  to  happen.  The  simplest  kind  of  trial  is  that  of  throw- 
ing a  coin  or  die.  At  each  throw,  any  side  has  an  opi)ortnnity 
to  come  up.  Then,  if  we  throw  50  pieces,  or  which  amounts 
to  the  same  thing,  throw  the  same  piece  50  times,  there  will 
be  50  trials;  and  we  may  inquire  into  the  probability  that  a 
given  side  will  be  thrown  exactly  9  times  in  these  trials. 

The  same  conception  occurs  in  another  form  if  we  have  50 
men,  each  of  whom  has  an  equal  chance  of  dying  within 
5  years.  Waiting  to  see  if  any  one  man  will  die  in  the  course 
of  the  5  years  is  a  trial.,  so  that  there  are  50  trials  in  all,  and 
we  may  inquire  into  the  probability  that  9  of  the  men  will  die 
during  the  trials,  just  as  in  the  case  of  50  throws  of  a  die. 

Let  us  distinguish  the  several  trials  by  the  letters 
a,  1),  c,  d,  €,  ....  w, 
which  must  be  n  in  number. 

1.  In  order  that  the  event  may  not  hap[>en  at  all,  it  must 
fail  on  every  one  of  the  n  trials.  The  probaljility  of  this 
(§  2G8,  Th.  II)  is  (1  —py  This  is  therefore  the  probability 
that  it  will  not  happen  at  all. 

Becaus'  the  probability  of  the  event  hapix-ning  on  any  one 
trial  is  p,  the  probability  of  its  failing  is  1  —  p.  AVe  now 
compare  the  possible  results. 

2.  The  event  may  happen  once  on  any  one  of  the  n  trials, 
a,  h,  r,  etc.  In  order  that  it  may  hapjien  only  once,  it  must 
fail  on  the  other  n  —  1  trials.  The  probability  that  it  will 
happen  on  any  one  trial,  say  e,  and  also  fail  on  the  remaining 
71  —  1  trials  is,  by  the  same  theorem, 

Because  there  arc  n  trials  on  which  it  may  equally  happen, 
the  probability  that  it  will  happen  once  and  only  once  is 

np{\.  —pY~^. 


■t  appVica- 
lons,  ospc- 
The  cou- 
)rebeiH\ocl. 
t  an  oppor- 
t  of  tln-oNV- 
,pportuuity 

5.  there  will 

uility  that  a 

trials. 

;  we  liave  50 

Iviiig  within 

(n  the  course 

vlsinalhand 

.  men  will  die 
of  a  die. 

tters 


at  all.  it  must 
.uhility  of  this 
[the  probability 


im, 

\-p. 


(T  on  any  one 


y\Q  now 


of  the  n  trials, 
fv  once,  it  nuist 
jiity  that  it  will 
[u  the  remaining 


I  ennally  happen, 
)uly  once  is 


1 


PROBABILITIES. 


313 


3.  The  event  may  liappcn  twice  on  any  two  trials  ont  of  tlio 
n  trial-.  In  order  tliat  it  may  happen  twice  only,  it  must  I'tiil 
on  the  other  ?i  —  2  trials.     Taking  any  one  combination,  say 

Happen  on    b,  d; 

Fail  on  a,  c,  e,  .  .  .  .  n, 

the  proljability  is  j!>^(l  —  pY~^. 

Bnt  it  may  happen  twice  on  any  combination  of  two  fri:ils 
ont  of  the  n  trials,  a,  h,  c,  .  .  .  .  n.  Because  these  coinJmia- 
tion."i  are  mutually  exclusive  (§  270),  the  total  ji'obability  of 
happening  twice  is 

4.  In  general,  in  order  that  the  event  may  happen  just  s 
times,  it  must  happen  on  some  combination  of  s  trials,  and  fail 
on  the  complf'nentary  combination  of  n  —  .v  trials.  The 
probabiiiiv  01  one  combination  is  ^Z  (1 — /?)'*"*  and  there 
are  C'l  .«uch  combinations.  Hence  the  general  probability  of 
hapi^eniug  s  times  is 

C'',  f  {I  -  p)n-s,  {a) 

If  there  is  on  each  trial  an  ecpial  chance  for  and  against 
the  event,  then  p  =  ^  and  l—p  =  --  The  probability  of 
the  event  happening  s  times  then  becomes 


This  ca.se  corresponds  to  that  already  treated  in 
Problem  2,  and  the  result  is  the  same  there  found. 


207, 


EXERCISES. 


I.  A  die  having  two  sides  wliite  and  four  sides  black  is 
thrown  5  times.     AVhat  are  the  respective  probaljilities  of  a 

white  side  Ijeing  thrown  1,  2,  3,  4,  and  5  times? 

* 

Note.    Here  p,  the  probability  of  a  white  side  on  one  throw,  is     ,  and 

o 
o 
1  — /)  =  3  "  Tl^*-'  ttuinber  n  of  trials  is  5, 


^h 


314 


PROBABILITIES. 


'I 


2.  Of  o  licalthy  men  figod  50,  the  probahilify  tliat  any  (  .n 
will  live  to  80  is     •    AVlnit  is  the  probability  that  three  or 

more  of  them  will  live  to  that  age? 

3.  A  chess-})layer  whose  chances  of  winning  any  one  game 
from  his  opponent  are  as  2  to  1,  undertakes  to  Avin  3  games 
out  of  4.     What  is  the  probability  that  he  will  be  able  to  do  it? 

Note.  It  would  be  a  fallacy  to  suppose  that  the  probability  required 
is  that  of  wiuning  exactly  3  games,  because  he  will  equally  wiu  if  he 
wins  all  four  games. 

272.  Events  of  Maxinmm  Prohability.  Returning  to  the 
general  expression  («),  let  us  inquire  what  number  of  times 
the  event  is  most  likely  to  occur  on  n  trial5.  The  rerpiireil 
number  is  that  value  of  s  for  which  the  probability 

is  the  greatest. 

If  we  call  P^  the  probability  that  the  event  will  happen 
exactly  .s  times,  and  if  s  is  to  be  the  number  for  which  the 
probability  is  greatest,  we  must  have 

Ps  >   Ps-X, 
Ps  >  Psil. 

Substituting  for  these  quantities  the  corresponding  forms 
of  the  expression  (a),  which  is  equal  to  Ps,  we  have 

C^  ;;*  (1  -  ;;)"-^  >  Ctip^'  (1  -  J»)«-*+S 
•     C^p'il  -pY-^  >  C?+i^«+i  (1  -  pY-^\ 
The  general  formula  for  C'«  in  §  257  gives 

s  +  1 


(*) 


m  = 


8 


C^- 


8~h 


(c) 


s  +  1 


Hence  we  have,  by  dividing  both  terms  of  the  first  in- 
equality C^*)  by  C«-i;/-i  (1 -i;)'^-*, 

n  —  s  + 1     .     . 
p  >  1—p. 


i 


4 


4 


PROBABTLITIES. 


315 


three  or 

one  game 
in  3  games 
le  to  do  it? 

ility  Toquired 
Uy  wiu  if  lie 

ning  to  the 
,er  of  times 
Che  required 


will  happen 
:or  whieli  the 


{c) 


of  the  first  in- 


Multiplying  by  s,  this  becomes 

np  —  ajj  ■{-  J)  y  s  —  sj). 
Iiitercliauglng  the  members  and  reducing,  we  have 

s<p{H  +  l).  {(l) 

Now  divide  the  second  inequality  (/>)  l)y  C'";;*  (1  — ^j)'*-^S 
and  reducing  by  the  second  equation  (t),  we  have 

.  ^  n  —  s 

i-?'>7+-i^- 

Multiplying  by  s  +  1  and  reducing,  we  find 

s  >;;(w-f  1)-1.  (e) 

Comparing  the  inequalities  (d)  and  (f),  we  sec  that  s  Ywa 
iK'tween  the  two  quantities  2>  ('-  +  1)  and  ;;  {ii  -\-  1)  —  1; 
that  is, 

s  is  tJie  £JTatest  whole  iiuniber  in  })  (w  +  1). 

If  the  number  of  trials  n  is  a  large  number,  and  p  is  a  small 
fraction,  -p  {n  +  1)  and  pu  will  dill'er  only  by  the  traction  ^j. 
We  shall  then  have,  very  nearly, 

s  =  pii. 
.  That  is : 

Thi:orem  I.  J7ic  most  prohalilc  niunher  of  times  that 
an  event  will  happen  on  a  great  nuinher  of  trials  is  the 
product  of  the  number  of  trials  by  the  probahilitij  on 
each  trial. 

Example.  If  a  life  insurance  company  has  GOOO  members, 
and  the  probability  that  each  member  will  live  one  year  is  on 

the  average  ^,  then  the  most  probable  number  of  deaths 

during  the  year  is  100. 

Rem.  It  must  not  be  supposed  that  in  this  case  the  num- 
ber of  deaths  is  likely  to  be  exactly  100,  but  only  that  they 
will  fall  somewhere  near  it. 

There  is  a  practical  rule  for  determining  what  deviation 
must  be  guarded  against,  the  demonstration  of  which  requires 
more  advanced  mathematical  methods  than  those  employed  in 
this  chapter.    It  is: 


!l 


y^] 


if 


'» 


316 


ruoBAiiiLrnEs. 


Tii noKEM  IT.  Drvirtflojis  from  the  iiiost  prohnhle  mnn- 
hrr  of  (IcatJiH,  cqital  to  the  s(/uarc  root  of  that  nuinhcr, 
will  he  of  frc(nu']it  occurrence. 

DevldtioiLs  ninelb  greater  than  tJiis  squfire  root  will 
he  of  iiifre(/iten.t  occurrence,  and  deviatlojis  more  than 
twice  as  great  will  be  rare. 

Examples.  In  a  conipiiny  of  wliicli  tlie  proVai)le  annual 
number  of  deaths  is  10,  tlic  actual  number  will  C()niinonly  I'all 
between  10—  VlO  and  10  +  VlO,  or  between  7  and  13.  It 
will  very  rarely  h.'ippen  that  the  number  of  deaths  is  as  small 
as  4  or  as  large  as  10. 

If  the  company  is  so  large  that  the  most  probable  number 
of  deaths  is  100,  the  actual  number  will  commonly  fall  betwceu 
100  —  VlOO  and  100  4-  VlOO,  or  between  90  and  110. 

If  the  most  jjrobable  number  of  deaths  is  1000,  the  actual 
number  Avill  commonly  range  between  i)G8  and  103^. 

We  noAV  sec  the  foUowincf  result  of  this  theorem: 

Tlie  greater  the  iiinnher  of  deaths  to  he  ex])eeted,  the 
greater  will  he  the  proludtle  deviatioii,  hat  the  less  will  he 
tlie  ratio  of  this  deviatioii  to  the  luhole  nmnher  of  deaths. 

Examples.  The  reductions  of  iha  cases  just  cited  are 
shown  as  follows : 


Expcctod  munbcr 
of  tleiiths. 

Probable 
deviation. 

Rali 
to  exi 

()  of  deviation 
lectud  iiumbor 

10 

3 

0.33 

100 

10 

0.10 

1000 

32 

0.03 

Application  to  Life  Insurance. 

2*73.  ^At  each  age  of  human  life  there  is  a  certain  proba- 
bility that  a  person  will  live  one  year.  This  probability  di- 
minishes as  the  person  advances  in  age. 

It  is  learned  from  observation,  on  the  principle  described  in 
the  preceding  section,  that  events  in  a  vast  number  of  trials 
are  likely  to  happen  a  number  of  times  equal  to  the  i)roduct  of 
their  probability  on  each  trial,  multiplied  by  the  number  of 
trials. 


PROUAniLITIES. 


317 


00 1  I  fill 
re  than 

le  tmnu!i\  ^ 
iioiily  t'ii^l 
d  1:5.     It 
3  as  siiKill 

0  number 
11  between 

110. 

the  actual 


nectcil,  fliG 
CSS  will  he 
of  dcalhs. 

cited  arc 


iation 
mber. 


2V 


tain  proba- 
obability  di- 


describcd  in 
nbcr  of  triali-i 
he  product  of 
c  number  of 


Tlicrcforo,  by  dividing  the  wliole  number  of  times  tlic  event 
has  hap[)ened  by  the  whole  number  of  trials,  the  (luotient  is 
the  most  probable  value  of  the  probal)ility  on  one  trial. 

Example.  If  we  take  50,000  people  at  the  age  of  25,  and 
record  how  many  of  them  are  alive  at  the  end  of  one  year,  this 
is  making  50,000  trials  whether  a  person  of  that  age  will  live 
one  year. 

If  4'J,G50  of  them  are  alive  at  the  end  of  the  year,  and  350 
arc  dead,  wc  would  conclude : 

Probability  of  living  one  year,    ....     0.003 
rrobability  of  dying  within  the  year,  .    .     0.007 

The  probability  for  all  ages  may  be  determined  by  taking  a 
great  nnmber  of  infant."^,  say  100,000,  and  counting  how  many 
die  in  each  year  until  all  arc  dead.  If  n  are  living  at  tlic  age 
y,  and  n'  at  the  age  y  -\-  \,  then   the  probability  of  dying 

within  one  year  after  the  age  y  will  be ,  and  that  of 

n' 


11  —  n 
n 


living  will  be 


n 


It  is  not,  however,  necessary  to  wait  through  a  lifetime  to 
reach  this  conclusion.  It  is  suflicient  to  find  from  o])servation 
what  proportion  of  the  i)eople  of  each  age  die  during  any  one 
year.  Suppose,  for  instance,  that  tiie  census  of  a  city  is  taken, 
and  it  is  found  that  there  are  ::250O  persons  aged  30,  and  2000 
aged  50.  At  the  end  of  a  year  another  in(|uiry  is  made  to 
ascertain  how  many  are  dead.  It  is  found  that  20  of  the  30 
year  old  people,  and  30  of  the  50  year  old  people  have  died. 
This  would  show: 

At  age  30,  probability  of  dying  within  1  year  =  0.008. 
*'       50,  "  "  "  ''  =  0.015. 

This  same  probability  being  obtained  for  every  year  of  life, 
the  probal)ility  of  living  1  year  at  all  ages  would  be  known. 
Then  a  table  of  mortality  could  bo  formed. 

A  table  of  mortality  starts  out  Avith  any  arbitrary  number 
of  people,  generally  100,000,  at  a  certain  age,  fre((uently  10 
years.  It  then  shows  how  many  of  these  people  will  be  living 
at  the  end  of  each  subse(iuent  year  until  all  are  dead.  The 
following  is  a  specimen  of  such  a  table. 


:il 


:1 


t  ■ 


I'^i 


318 


PliOBADILITIES. 


Table  of  Mortiility. 


•» 


Prob,  of 

Prob.  of 
dying 
Willi  n 

Prol).  of 

Prob  of 

Ages. 

Living. 

Dying. 

Hiirviving 

Ages. 

Living. 

Dying.  Hiirviving 

living 
williiii 

a  year. 

tlio  year. 
.00442 

60 

.677 

a  year. 

.97127 
.9689^ 

the  year. 
.02H72 

ir 

100000 

442 

•99558 

58373 

II 

qq55S 

407 

385 

.99591 

.00408 

61 

56696 

1760 

.o3i()4 

13 

991 5i 

.996 1 1 

.00388 

62 

54936 

|8.',9 

.966)4 

.o3365 

i3 

98766 

376 

.99619 

.oo38o 

63 

53087 

1936 

.96353 

.03646 

14 

98390 

379 

.99614 

.00385  1 

64 

5ii5i 

2014 

.96062 

.03937 

i5 

9801 1 

396 

.99595 

.00404  ; 

65 

49137 

2080 

.95766 

.04233 

i6 

97615 

426 

.99563 

.00436 

66 

47057 

21 38 

.95456 

.04543 

>7 

97189 

469 

.99517 

.00482 

67 

44919 
42733 

2186 

.95133 

.04^66 

l8 

96720 

523 

.99457 

.00542  ' 

68 

2224 

.9479^ 

.o52o4 

»9 

96195 

58 1 

.99396 

.oo6o3 

69 

4o5og 

2268 

,94401 

.05598 

20 

95614 

621 

.99350 

.00649 

70 

38241 

233 1 

.93904 

.06095 

21 

94993 

645 

.99321 

.00679 

7' 

35910 

2401 

.93313 

.066H6 

22 

94348 

653 

.99J07 

.00692 

72 

33509 

2469 

.92631 

.07368 

2] 

93695 

65i 

.993o5 

.00694 

73 

3 1040 

253 1 

.91846 

.08154 

24 

95044 

647 

.99304 

,00695 

74 

285o9 

2567 

.90995 

.09004 

25 

92397 

647 

.99299 

.00700 

75 

25942 

2542 

.90201 

.09798 

26 

9i7:)o 

65i 

.99290 

.00709 

76 

23400 

247''' 

.89418 

.io58i 

27 

91099 

668 

.99266 

.00733 

]l 

20924 

2369 

.88078 

.11321 

28 

90431 

686 

.99241 

.00753 

1 8555 

2247 

.87S90 

,12109 

29 

89745 

703 

.99216 

.00783 

79 

i63o8 

2110 

.87061 

.12938 

3o 

89042 

718 

.99193 

.00806 

80 

14198 

1969 

.861 3 1 

.13868 

3i 

88324 

726 

.99178 

.00821 

81 

12229 

1823 

.85092 

,14907 

32 

87598 

733 

.99163 

.00836 

82 

10406 

1672 

.83932 

.16067 

33 

86865 

743 

.99144 

.00855 

83 

8734 

l522 

.82373 

.17426 

■34 

86122 

754 

.99124 

.00875 

84 

7212 

i36o 

.81142 

.18857 

35 

85368 

768 

.99  TOO 

.00899 

85 

5852 

1186 

.79733 

.20266 

36 

84600 

789 

.99067 

.00932 

86 

4666 

1014 

.78268 

.21731 

37 

838II 

811 

.99032 

.00967 

87 

3652 

849 

.76752 

.23247 

38 

83ooo 

83o 

.99000 

.01000 

88 

2803 

689 

.75419 

.24580 

39 

82170 

844 

.98972 

.01027 

89 

2114 

548 

.74077 

.25922 

40 

8i326 

854 

.98949 

.oio5o 

90 

1 566 

435 

.72222 

•27777 

41 

80472 

860 

,98931 

,01068 

9« 

n3i 

336 

.70291 

.29708 

42 

79612 

869 
888 

.98908 

.oiogi 

92 

It 

247 

.68930 

.3 1 069 

43 

78743 

.98872 

.01127 

93 

181 

.66970 

.33029 

44 

77855 

9i3 

.98827 

.01172 

94 

367 

i3i 

.643o5 

.35694 

45 

76942 

948 

.98767 

.01232 

95 

236 

86 

.63559 

.36440 

46 

7^994 

989 

.98698 

.oi3oi 

96 

i5o 

56 

.62666 

.37333 

47 

75oo5 

1029 

.98628 

.01371 

97 

94 

44 

.53191 

.46808 

48 

7397b 

1067 

.9S557 

.01442 

98 

5o 

33 

.34000 

.66000 

49 

72909 

1102 

.98488 

.oi5i  I 

99 

>7 

II 

% 

?1 

5o 

71807 

ii33 

.98422 

.01577 

100 

6 

4 

H 

% 

5i 

70674 

1 167 

.98348 

.01 65 1 

101 

2 

2 

52 

69307 

1204 

.98267 

.01732 

102 

0 

.... 

53 
54 

683o3 
67052 

125l 

i3o4 

.98168 
.98055 

.oi83i 
.01944 

Note.  Tlie  above  table  is 

that  of 

55 

65748 

1 358 

.97934 

.02065 

the 

Enixlish  Institnto  of  Act 

iiarics, 

56 

64390 

1414 

.97S()4 

.02195 

pre 

pared  between  IHfW  and  IStii 

),  from 

57 
58 
59 

62976 
6i5o5 
59974 

1471 
i53i 
1601 

.97664 
.97510 
.97330 

.02335 
.02489 
.02669 

tlie 
lea 

continued  experience  of  1 
ding  life  inBurance  compau 

.wenty 
ics. 

PROnABILITIES. 


319 


,  '  Prob  of 

I  dvliit,' 
'«  within 
■     the  J  car. 


) 

l 
J 
2 

5 
f) 
3 
5 
I 

i4 
3 
>i 
i6 
)5 

31 

iH 

78 
90 
61 

3i 

)2 

73 
42 

|;33 

8 

132 
•9 

f)77 

22 
2Q1 
c;3o 

97" 
3o3 

191 
000 


K 


,02872 
.o3io4 
.o33()5 
.o36.U) 
.03937 

.04233 
.o4:).',3 

.0/,^^()() 

.o!)2().i 
.o5d9H 

.06095 
.o6fiKf> 
.0736H 
.08  if).', 
.09004  I 

.09798 
.iofi8i  I 

.Il32I 

.12109 
.12938 

.i3%8 

.i/.9(>7 
.i()o()7 

.17.126 
.18837 

.20266 
.21731 

.23247 
.24580 

.2:)922 

.27777 

.2970B 
.3 1 069 
.33029 
.35694 

.36440 
.37333 
.46808 
.66noo 
».; 


ble  if  that  of 
Actnaricn, 
lul  1809,  from 
;e  of  twenty 
»mpauieB. 


Phoijlkm.  To  find  tlie  i)robubility  that  ti  person  of  ago  a 
will  live  to  uge  ?/. 

Sulution.  Wo  take  from  the  tahle  the  mniiher  living,'  at 
age  ?/,  and  divide  it  hy  the  number  living  at  age  a.  The  (^uo- 
tient  is  the  probability. 

274.  The  principle  on  which  the  value  of  a  contingent 
payment  is  determined  is  the  following : 

Theorem.  Tlic  value  of  ct  pvohahlc  pdiimrnt  ift  equal 
to  the  siuii  to  he  paid,  viidti plied  by  the  proffafji/ifi/  that 
it  will  be  paid. 

Proof.  Let  there  bo  w  men,  for  each  of  whom  there  is  a 
])robability /)  that  he  will  receive  the  sum  s.  Then  l)y  §  2'i'Z, 
Th.  I,  pn  of  the  men  will  probably  receive  tiie  i)aymont,  so  that 
the  total  sum  which  all  will  receive  will  i)robably  bo  pus.  Now, 
before  they  know  who  is  to  get  the  money,  the  value  of  each 
one's  share  is  ecpial.  Therefore,  to  find  this  value,  we  divide 
the  whole  amount  to  be  received,  namely,  /jwn,  by  the  number 
of  men,  n.  This  gives  ps  as  the  value  of  each  one's  chance, 
which  proves  the  theorem. 

Note.  In  this  proof  it  is  tacitly  supposed  that  the  pus 
dollars  arc  as  valuable  divided  among  the  p)i  men  as  divided 
among  all  7i  men.  But  this,  though  supposed  in  mathematical 
theory,  is  not  morally  true.  Morally,  the  money  will  do  more 
good  when  divided  among  all  the  men  than  when  divided 
among  a  portion  selected  by  chance.  All  gambling,  whether 
by  lotteries  or  games  of  chance,  is  in  its  total  effects  upon  the 
pecuniary  interests  of  all  parties  a  source  of  positive  disadvan- 
tage. This  disadvantage  is  treated  mathematically  by  more 
advanced  methods  in  special  treatises. 

EXEFICISES. 

I.  Find  from  the  table  the  probabilities  that  a  person 

a.  Aged    30  will  live   to   70. 

b.  "       30     "  "       80. 

c.  "       50     "  "       60. 

d.  "       60     "  "       70. 


V\ 


h 


•liH 


|r 


n20 


puonxjiiLirrEf^. 


'I 


c. 

/. 

h. 


A^ccl    ro    will    live    to   HO. 


tt 


80 

» t 

1 1 

00. 

00 

<( 

<( 

05. 

95 

(( 

(( 

100. 

2.  \VI)jil  n^e  is  tlitit  at  wliich  it  is  an  even  chance  wiictlier 
u  ])t'rs()ii  n^'ctl  40  will  he  liviii;,^  <»r  tlcad  ? 

3.  Show  that  the  i)rohal)ility  that  a  person  nj^ed  'M)  will  live 
to  TO  is  C(|ual  to  the  ])ro(lut't  of  tlie  i)rohMl)ility  that  lie  will  live 
to  00  imiltii>lic'il  by  the  })rol)abili(y  that  a  man  aged  GO  will 
live  to  To.     (Apply  the  thooreni  of  §  ;.'(;0.) 

4.  What  pi'cininni  onu^ht  a  man  of  <i5  to  pay  for  insnring 
his  life  for  Ji^^OOO  for  I  year? 

5.  Ten  younf]^  men  of  -.'5  form  a  elnb.  What  is  the  proba- 
bility that  it  will  \w.  nnbroken  by  death  for  ten  years  ? 

6.  The  probability  that  a  planing  mill  will   bnrn   down 

within  any  one  year  is  ;.•     What  onght  an  insnranee  company 

») 

to  charge  to  insnrc  it  to  the  amonnt  of  ^3000  for  1  year,  for 
2  years,  for  3  yeai'S,  and  for  4  years,  res])eetivt.'ly  ? 

7.  If  the  probability  that  a  honse  w  ill  burn  down  in  any 

one  year  is  ;^,  what  onght  to  be  the  premium  for  insuring  it 

for  6'  years  to  the  amount  of  a  dollars? 

NoTio.  In  cnst'.s  like  the  last  two,  it  Ih  assumcil  that  only  one  loss 
will  bo  paid  for. 

8.  What  is  the  probability  that  if  a  man  aged  25  marry  a 
"wife  of  20,  they  will  live  to  celebrate  their  golden  AVedding? 

9.  A  comi)anv  insures  the  joint  lives  of  a  husltand  aged  70 
and  a  wife  aged  50  for  85000  for  5  years,  the  stipulation  being 
that  if  either  of  them  die  within  that  time  the  other  shall  be 
paid  the  money.  What  ought  to  be  the  premium,  no  allow- 
ance being  made  for  interest  ? 

10.  A  man  aged  50  insures  the  life  of  his  Avife,  aged  35,  for 

$10,000  for  20  years,  with  the  promise  that  the  money  is  not 

to  be  paid  unless  he  himself  lives  to  the  age  of  70.     What 

onght  to  bo  the  ])reniium? 

Note.  In  computations  rclatinjx  to  the  inana<]fonicnt  of  life  insurance, 
it  is  always  necessary  to  allow  compound  interest  on  all  jjayments.  But 
the  above  exercises  are  intended  only  to  illustrate  the  application  of  the 
theory  of  jirobaliilities  to  tlie  suliject,  and  therefore  uo  allowance  for  in- 
terest ia  expected  to  be  made  in  the  answers. 


Mi 


!  Avliether 

)  will  live 
D  will  livo 
id  OU  will 

L*  insuring 

lie  prol);i- 

urn  down 
3  com  puny 
1  your,  fur 

)wn  in  any 
insuring  it 

inly  one  loss 

•^5  marry  a 
c'ddingV 
nd  aged  TO 
ation  being 
ler  shall  be 
11,  no  alluw- 

agcd  35,  for 
Loney  is  not 
"  70.    What 


life  insurance, 
lymtmts.  But 
licutinu  of  the 
iwanco  for  in- 


BOOK    XI. 

OF  SERIFS  ANP  TlfF  DOCTRINE   OF 

LIMITS. 


CHAPTER    I. 

NATUREOFA     SERIES. 

fi7i>.  I>(f.  A  Series  is  a  succession  of  tci-nis  follow- 
ing L'licli  other  accoidiug-  to  some  goucral  law. 

ExAMi'LKs.  An  arithmetical  jirogression  is  a  series  dot er- 
mined  by  the  law  tiiat  each  term  shall  be  greater  than  the 
preceding  one  bv  the  same  amount. 

A  geometrical  ])rogro^•sion  is  a  series  subject  to  the  law 
that  the  ratio  of  evory  two  consecutive  terms  is  the  same. 

These  two  progressions  are  the  sim[)lest  form  of  series. 

A  series  may  terminate  ut  some  term,  or  it  may  continue 
indefinitely. 

Def.  A  scries  which  continues  indeliuitely  is  called 
an  Infinite  Series. 

Dcf.  The  Sum  of  a  series  is  the  algebraic  sum  of 
all  its  terms.  Hence  the  sum  or  an  inlinit(^  series  will 
consist  of  the  sum  of  an  inlinite  number  of  terms. 

3T0.  The  law  of  a  series  is  generally  such  that  the  n^^ 
term  may  be  expressed  as  a  function  of  n. 
For  example,  in  the  series 


1111 

^  +  3  +  4  +  5  +  '^'•' 


the  n^^  term  is 
21 


w  +  1 


,rf 


SERIES. 


'» 


,    I 


T        .1  •  111. 

In  tlie  senes      --  +  -—  +  -~  +  etc., 
the  n^^  term  is 


n  {u  +  1) 

D(f.     The  expression  for  the  ;?'*  term  of  a  series  as 
a  function  of  /i  is  called  the  General  Term  of  the ' 
series.  » 

EXERCISES. 

Express  the  n^^  teiii  of  each  of  tiie  folio  whig  series  : 

111. 

'•     T~r  +  T~  +  FT  +  ^^^• 

O'-i         -i-o  O'b 

2.     1-2  +  3.4  +  r-G  +  etc. 

3-  ^  +  r2  + 1:2^ 3  +  ^^'- 

«  fr^  ^j4  ,j8 

4-     2.-2  +  3:22  +  4^23  +  5T24  +  etc. 

Write  four  tenns  of  each  of  the  series  having  the  following 
general  terms: 

5.  The  71^^  teriii    0  l-o   ,  .,      -• 

6.  The  i'^  term  to  ^  ?.  i  {i  +  1)  {i  +  2)  2^. 

7.  The  {)i  +  1)*'  term  to  be  ^  ^  ^  ' 


{n  +  5)  («  +  G) 


8.  The  (/?  —  1)*'  term  to  he  — - 


^/»-l 


1-2  ....  w 

277.  The  most  common  nse  of  a  series  is  to  enahle  ns  to 
compute,  l)y  ai)proxiniation,  the  values  of  expressions  wliieh  it 
is  ditlieult  or  impossible  to  compute   directly.     Sui)pose,  for 

exami)k',  that  Ave  have  to  compute  the  value  of '-  when  x 

is  a  small  fraction,  say  ^^,  and  to  have  the  result  accurate  to 

eight  decimals.     We  shall  see  hereafter  that  when  x  is  less  than 
1,  we  have 


f 


CONVERGENCE  OF  SERIES. 


323 


1  -\-x 
1—x 


=  1  +  22:  +  2^:2  4-  22-3  +  etc.,  ad  infinitum. 


Suppose  X  =z  ~-  z=  .02. 


2  X  .02  = 

Multii)lying  by  .02, 


We  comjmte  this  series  thus : 

1 

.04 

.0008 
.00001 G 
.00000032 


1.02 


^^^"^  =  ms  = 


1.04081(.i32 


which  IS  much  more  expeditious  than  dividing  1  02  hy  .98. 

It  will  be  seen  that  every  term  we  add  makes  the  (luotient 
accurate  to  one  or  two  more  decimals,  so  that  there  is  no  limit 
to  the  precision  which  may  be  attained  by  the  use  of  the  series. 

If,  however,  .r  had  been  greater  than  unity,  the  series  would 
give  no  result,  because  the  terms  2.r,  'Zx^,  'Zx^,  would  have  gone 
on  increasing  indefinitely,  whereas  the  true  value  of  the  frac- 

\  -\-  X 

tion  — would  have  been  negative. 

1  —  X  ° 

This  example  illustrates  the  following  two  cases  of  series : 

I.  There  mcnj  be  a  certain  limit  to  irhieli  the  sinnof 
the  series  sJialt  aj)])roaeh,  as  ire  increase  the  inimher  of 
terms,  but  which  it  can  never  reach,  how  great  soever  the 
number  of  terms  added. 

For  example,  the  series  wo  have  just  tried, 

1.02 
approaches  the  limit  ;;-f,",  but  never  absolutely  reaches  it. 

0.  J8 

II.  As  we  increase  the  nuniher  of  terms,  the  sunv 
maij  increase  without  limit,  or  majf  vil)rate  l/ac/c  and 
forth  in  consequence  of  some  terms  being  positive  and 
others  negative. 

These  two  classes  of  series  are  distinguished  as  conceryent 
and  divenjcnt. 


h 


324 


SERIES. 


\i  I 


\  {•' 


\  i 


M 


Dpf.  A  Convergent  Series  is  one  of  whicli  tho  sura 
approaches  a  limit  as  the  nuiuber  of  terms  is  increased. 

liefer  to  §  213  for  an  example  of  infinite  series  in  geometrical  pro- 
grcsr^ions  which  have  limits. 

Drf.  A  Divergent  Series  is  one  of  wliich  the  sum 
does  not  approach  a  limit. 

Examples.  The  series  l  +  ^-f  3  +  4  +  ctc.,  ad  lufnutum, 
is  divergent,  because  there  is  no  Hniit  to  the  sum  of  its  terms. 

The  series  1  —  1-f  1  —  1  +  1  —  etc.,  is  divergent,  because 
its  sum  continually  fluctuates  between  +1  and  0. 

Rem.  When  we  consider  only  a  limited  nnmlier  of  terms, 
the  (|nestion  of  convergence  or  divergence  is  not  important. 
]5ut  when  the  sum  of  tlie  whole  series  to  infinity  is  to  be  con- 
sidered, only  convergent  series  can  be  used. 

Notation  of  Sums. 

274.  The  sum  of  a  series  of  tenns  represented  by 
common  symbols  may  be  expressed  by  the  symbol  2, 
followed  by  one  of  the  terms. 

Example.     The  expression 

moans  "  the  sum  of  several  terms,  each  represented  by  «." 

When  it  is  necessary  to  dlstinguisli  the  different 
tenus,  dilFerent  acc(Mits  or  indices  are  affixed  to  thein, 
and  represented  by  some  common  symbol. 

Example.     The  expression 

means  the  snm  of  several  terms  represented  by  the  symbol  a 
with  indices  attached  ;  that  is,  the  sum  of  several  of  the  '|uan- 
tities  a^,  (1^,  a.^,  a^,  etc. 

When  the  particular  indices  included  in  the  summa- 
tion are  to  be  expressed,  the  irieatt'st  and  least  of  them 
are  written  above  and  below  the  symbol  i. 


1 

i 


fti 


SIGJV   OF  SUMMATION, 


325 


;lio  sum 
:reasf(l. 

trical  pro- 
lie  sum 

finitum, 
s  terms, 
because 

f  terms, 
portiiiit. 
)  be  cou- 


ntod  by 
iibol  2, 


a. 


» 


liff(U'cnt 
0  tliem, 


symbol  a 
lie  '|uan- 

summa- 
ot'  tliem 


Examples.     The  expression 


i=5 


means :  "  Sum  of  all  the  syml)ols  ai  formed  by  giving  i  all  in- 
tegi-al  values  from  «  =  5  to  i  =  15."     That  is, 


:15 


lai  =  ai  +  «6  +  a,  +  a,  +  r^  +  rt,o  +  ff„  +  f/,.;  +  r/„+^,4  +  r^5. 


i^5 


lim  means  0  +  m  +  2m  +  3w  -f  4?«  +  5m. 


i=0 

^1  (/,./)  means  (1,.;)  +  {'^,J)  +  (3,;)  +  (4,^). 
yihj)  =  {i,  2)  +  {i,  3)  +  {i,  4)  +  (/,  5)  +  (/,  G). 
i;;i!  =  l!  +  2!  +  3!  +  4!  =  1  +  2  + G  +  24  =  33. 

n=l 
i=li 

1/  =  7  +  8  +  9  +  10  +  11  =  45. 

«-=7 
j— 5 

"^^i-a  =  2^2  +  32  +  42  +  52  =  54. 

EXERCISES. 

Write  out  the  following  summations,  and  compute  their 
values  wlien  they  are  purely  numerical : 


I.       ^/K 
4.       llmi. 

i=4 
i=4 

7.       It  mi. 


n=6  n=6 

2.         5;/^(m_1).         3.         2?i(;2  +  l). 


n=7 

n=4 

?i=S 

n-2 


n  =  l 

6.     ""2  0^  +  1)0-1). 
n-0 


i^'-J  «^a  n=o  ^i  +  1 

Express  tlie  following  sums  by  the  sign  1: 
10.     //o+^'i+Z's+Z^a+Z/^.  II.     13  +  23  +  33  +  43. 

12.     1-2  +  2-3  +  3-4  +  4 


5. 


12       3       4       5 

13.   o  +  Q  +  A  +  n +  ,r 


Hi 


2    '   3    '   4    ■   5    '    G 


326 


SEiiirs. 


M 


CHAPTER     li. 

DEVELOPMENT    IN    POWERS    OF    A    VARIABLE. 

270.  Among  the  most  common  series  employetl  in  nititli- 
ematics  are  those  of  which  tlie  terms  are  multiitlied  by  tlie 
successive  powers  of  some  one  quantity. 

An  examine  of  such  a  series  is 

1  +  2z  +  3^2  +  4^3  +  or*  4-  etc., 

in  wliich  each  coefficient  is  greater  by  unity  than  the  power  of 
z  which  it  multiplies. 

A  geometrical  progression,  it  will  be  remarked,  is  a  scric 
of  this  kind,  in  which  the  terms  contain  the  successive  powers 
of  the  common  7'atio. 

The  general  form  of  such  a  scries  is 

Oq  +  a^z  +  a^z^  +  «32^  +  etc., 

in  which  the  successive  coefficients  a^,  a^,  a^,  etc.,  are  formed 
according  to  some  law,  but  do  not  contain  z. 

Such  a  series  as  this  is  said  to  proceed  according  to  the 
ascending  powers  of  the  variable  z. 

Rem.  The  sum  of  a  series  is  often  equal  to  some  algebraic 
expression  containing  the  variable.  Conversely,  we  may  find  a 
series  the  sum  of  all  the  terms  of  which  sliall  Ije  equal  to  a 
given  expression. 

Def.  A  series  equal  to  a  given  expression  is  called 
the  Development  of  that  expression. 

To  Develop  an  expression  means  to  find  a  series 
the  sum  of  all  the  terms  of  wliicli  are  equal  to  the  ex- 
pression. 

The  most  extensively  used  method  of  development  is  tliat 
of  indeterminate  coefficients. 


i 


INDETERMINA  TE   COEFFICIENTS. 


327 


3LE. 

in  uiallj- 
d  by  the 


power  of 

s  a  scric 
e  powers 


e  formed 
ug  to  the 

algebraic 
my  find  a 
rjual  to  a 

is  called 

a  series 
^  the  ex- 

u  is  tliat 


3Ietliod  of  Iiideterniinate  Coefficients, 

2SO.  The  method  of  indeterminate  coefficients  is  based 
npon  the  following  pnnciples  : 

Let  lis  have  two  equal  expressions,  each  containing  a  varia- 
ble z,  and  one  or  both  containing  also  certain  indeterminalc 
quantitieSj  that  is,  quantities  introduced  hy])othetically,  and  not 
given  by  the  original  problem,  the  values  of  which  are  to  be 
subsequently  assigned  so  as  to  fulfil  a  certain  condition. 

The  condition  to  be  fulfilled  by  the  values  of  the  inde- 
terminate quantities  is  that  the  two  expressions  containing  z 
and  these  quantities  shall  be  made  identically  equal. 

Thei>,  Vxicause  the  equations  arc  to  be  identically  equal,  we 
can  asai^  any  values  we  please  to  z,  and  thus  form  as  many 
equations  as  we  please  between  the  indeterminate  quantities. 

If  these  equations  can  be  all  satisfied  by  one  set  of  values  of 
these  quantities,  then  by  assigning  these  values  to  them  in  the 
original  equation,  the  latter  will  be  an  identical  one,  as  required. 

The  stndfnt  pbould  trace  the  above  general  method  in  the  following 
examples  of  its  application. 

281.  Theorem  I.  If  a  scries  proceeding  according 
to  the  ascending  powers  of  a  quantity  is  equal  to  zero  for 
all  values  of  that  qnantitij,  the  coeflicicnt  of  each  sepa- 
rate term  must  he  zero. 

Proof.  Let  the  several  coefficients  be  a^,  f/,,  a^,  etc.,  and 
2  the  quantity,  so  that  the  series,  put  equal  to  zero,  is 

«r„  +  fl'jZ  +  ac,7?'  4-  a^z^  +  etc.  =  0. 

Becanse  t'lC  equation  is  true  for  all  values  of  z,  it  must  be 
true  when  z  =  0.     Putting  z  =:  0,  it  becomes 

it^  —  0. 
Dropping  a^,  the  equation  becomes 

ff,2  -f  a.^z^  +  a^z-^  +  etc.  =  0. 
Dividing  by  z,  a^  +  n„z  +  a./^  +  etc.  =  0. 
From  thii*  we  derive,  by  a  repetition  of  the  same  reasoning, 

«i  =  0. 


II 


■  f  I 
■\\ 

ill 


!» 


328 


SERIES. 


•I 


Continuing  the  process,  avc  find 

fto  ~  0,     fl'a  =  0,     etc.,  indefinitely. 

TnEOREM  II.  //  two  series  jn'occedhig  by  ascending 
poircrs  of  a  quantity  are  eqitrd  fo/'  all  values  of  tJiat 
quantity,  the  coejjiclents  of  tlio  Cfjual  powers  must  he 
equal. 

Proof.     Let  the  two  equal  series  be 

aQ-\-aiZ-\-a,^z^-\-Qt(i.  ~  h(^-{-h^z-\-h2Z^-\-Qtc.  (a) 

Transposing  the  second  member  to  the  left-hand  side  and 
collecting  the  equal  powers  of  z,  the  equation  becomes 

^'o  ~  ''''o  +  (^'i  —  ^i)  ^  +  i'^'z  —  ^s)  ^^  +  etc.  =  0. 

Since  this  e([uation  is  to  be  satisfied  for  all  values  of  z,  the 
coefficients  of  the  separate  powers  of  2  must  all  be  zero. 

Hence, 

^^0  ~  ^''o  =  0,        rtj  —  ^1  =  f*j        ^8  ~  ^3  —  0,       etc. 
or  Hq  =  h^,  a^  =  b^,  a^  =  b^,     etc. 

Exercise.  Let  the  student  demonstrate  these  last  equa- 
tions independently  from  (a),  by  sujjposing  z  ^=0,  then  sub- 
tracting from  botli  sides  of  (a)  the  (juantities  found  to  be  equal; 
then  dividing  by  z  ;  then  supposing  z  =:  0^  etc. 

Rem.  The  hypothesis  that  (a)  is  satisfied  for  all  values  of 
z  is  equivalent  to  the  supposition  that  it  is  an  idenlical  equa- 
tion. In  general,  wlicn  we  lind  different  expressions  for  the 
same  functions  of  a  variable  (piantity,  these  expressions  ought 
to  be  identically  e([ual,  because  they  are  expected  to  be  true 
for  all  values  of  the  variable.  , 

Tiieoijem  III.  ,1  function  of  avariahle  can  oidyhc. 
drirl()])ed  in  a  single  way  in  ascending  powers  of  the 
varitd)1e. 

For  if  we  should  have 

Fz  z=z  A^-^  A ^z  -\-  A^z^  +  A^z?  +  etc., 
and  uUo      Fz  —  i/„  +  B^z  -\-  B^z^  +  B.^z^  +  etc., 


INDETERMINA  TE  COEFFICIENTS. 


329 


tscending 
?s  of  that 
i  mast  ho 


;c.  ('0 

nd  side  iind 
lies 

=  0. 

lies  of  z,  the 
zero. 


=  bo, 


etc. 
etc. 


sc  last  c(iua- 
0,  then  sub- 
to  be  equal ; 

all  values  of 

'leu/ical  vquci- 

!sioiis  for  the 

■ssions  ouj-ht 

id  to  be  true 

can  ovhj  ho 
oirers  of  the 


itc, 
itc, 


those  two  series,  being  each  identically  equal  to  Fz,  must  1)0 
identically  equal  to  each  other.  But,  by  Th.  II,  this  canuot  bo 
the  case  unless  we  have 

yl(,  =  />'(,,    ylj  =  7?  J,     ylo  =  B„y    etc. 

The  coefficients  being  equal,  the  two  series  are  really  one 
and  the  same. 


28^.  Expansion  hy  Indeterminate  Cocjjiricnts.  The  above 
principle  is  ap[)lied  to  the  devcl(i])ment  of  functions  in  powers 
of  the  variable.  The  method  of  doing  this  will  be  best  seen 
by  an  example. 

1.  Develop in  powers  oi  x. 

Let  us  call  the  coefficients  of  tlie  powers  of  x  a^,  a^,  etc. 
Tlie  series  will  be  known  as  soon  as  these  coefficients  uro 
known.    Let  us  then  suppose 


^0  +  0\^  +  ^2^^  +  ^3^^  +  etc. 


1  +  x 

Here  we  remark  that,  so  far  as  we  have  shown,  tliis  equa- 
tion is  purely  hypothetical.  We  have  not  proved  that  any 
such  equation  is  possible,  and  tlic  ([uestion  wliether  it  is  possi- 
ble must  remain  open  for  the  present.  We  must  find  whether 
we  can  assign  such  values  to  the  indeterminate  coefficients,  a^, 
«!,  «3,  etc.,  that  the  equation  shall  be  identically  true. 

Assuming  the  equation  to  be  true,  we  multiply  both  sides 
by  1  4-  X.     It  then  becomes 

1  =  ^^0  +  («o  +  «i)  ^  +  (^^1  +  ^s)  ^^  +  etc. ; 
or  transposing  1, 
0  =  «o  —  1  +  {aQ-\-a^)x  4-  {a^  -]-ac,)x^  -f-  {a„-\-a^)x?  -[-  etc. 

By  Theorem  I,  the  coefficients  must  be  identically  zero. 
Hence, 

«o  —  1    =0,    which  gives    «„  =  1 
«i  +  ^'o  =  0, 


«2  +  f?i  ==  0, 
«3  +  «s  =  0, 
etc. 


u 
n 
a 


« 


■0   —   "  ) 

«i  =  -^^  =  —  1; 

«2  =  — «i  =  1; 


«3    = 


«2  =  -1; 


11 


.  f 


(lii 


I  i 


etc. 


330 

SERIES. 

Substituting  these  ^ 

alues  of  the  coefficients 

ill 

the 

original 

oquatiun, 

it  becomes 

l+x~^ 

—  X  ■\-  x^  — 

7?  +  ^  — 

etc. 

This  same  metliod  can  be  applied  to  the  development  of 
any  rational  fraction  of  A\hich  the  terms  are  entire  functions 
of  some  one  quantity.     Let  us,  for  instance,  suppose 

m  +  nX  +  px^  0  -T-       1      -r       2       -r  -r       n 

Multii)lying  by  the  denominator  of  the  fraction,  this  equa- 
tion gives 

a  -\-  bx  =  iuAq  +  (n^o  +  mJi)a;  -f  {pAQ~{-7iA^-{-)nA„)x'^ 

+  (pA  1  +  uA  g  +  viA  3)  x^  -f-  etc. 

We  now  see  that  when  t  >  1,  the  coetricient  of  x*  in  this 
equation  is  inAi  +  JiAi^.i  ■}-  pAi^z. 

E(iuating  the  coefficients  of  Hke  powers  oi  x, 

a 


thAq  ■=  a,    whence    A^  =z 


m 


mA  ^  ■{■  iiAq  =.  h, 
mA^  +  nA^  -V pA^  =  0, 
mA^  +  iiAz  -\-pA^  =  0, 


C( 


(( 


(( 


A.  = Aq  ; 


I    -  -Ia   --a  ' 


—  R 


71 


^  7)1       *  7)1      ^ 


We  have  from  the  general  coefficient  above  written,  when 


A,= 


^^'     A  V     i 

—  Ai-t Ai-z. 

7)1  7)1 


That  is,  each  coefficient  after  the  second  is  the  same 
linear  function  of  tlie  two  coefficients  next  preceding. 

Such  a  series  is  called  a  Recurring  Series. 

EXERCISES. 

Develop  by  indeterminate  coefficients: 
1  1 


I. 


1—x 


2. 


1  —  22; 


c  oi'itrinal 


apment  of 
I'unctioiis 


tliis  equa- 

mAz)x^ 
3)  x^  4-  etc. 

i  x^  in  this 


UNDETERMINED   M UL  TIPLIER8. 


331 


ni 


^0; 


--A^'y 

m 
kttcn,  when 


the  same 
^ceding. 


3- 

5- 

7. 


1-x 
1  +  x 
1  +  a; 


1  -{-•ix+  'Sx^ 

1  +  'ZuT^dl^' 


4. 
6. 

8. 


1  +  2: 

1-r 

1  _  ;i^'  _|_  x^' 
1-X 


1  +  ^'  —  .-^ 


283.  The  development  of  a  rational  fraction  may  also  bo 
cITccted  by  division,  after  the  manner  of  §§  OG,  07,  the  oi)era- 
tion  being  carried  forward  to  any  extent. 

1  4-  aj 
Example.     Develop  « 

J.    ^"~    U/ 

1  -\-  X  \   1  —  X 

1-X 


1  +  2x  +  2x^  +  2xr^  -\-  etc. 


2x 

2x  —  2^ 


2x'^  +  0 
2x^  —  2x^ 


2x^,  etc. 


EXERCISES. 


Develop  by  division  the  expressions : 

1—2:?:  1  +  X 


I. 


1+a;  1  —  X  -{-  x^ 

284.  Elimination  hj  Undcterinined  3hiltipJiers.    There  is 

an  api)lication  of  the  method  of  undetermined  coefficients  to 

the  problem  of  eliminating  unknown  quantities,  which  merits 

special  attention  on  account  of  its  instructiveness.     Let  any 

system   of  simultaneous  equations  between   three  unknown 

quantities  be 

ax  -\-    hy  -\-    cz  =^  hy  (1) 

a'x  +  h'lj  +  c'z  =  h'y  (2) 

a"x  +  b"y  +  c"z  =z  h".  (3) 

Can  we  find  two  such  factors  that,  if  we  multiply  two  of 
the  equations  by  them,  and  add  the  results  to  the  third,  two  of 
the  three  unknown  quantities  shall  be  eliminated? 


III! 


'I 


h 


(I 


^M 


332 


SERIES. 


This  question  is  answered  in  Iho  followiiif^  way: 

ir  there  be  siieii  factors,  let  us  call  Ihcni  in  and  n.     If  wo 

niuUii)Iy  the  first  e(|uation  hy  in,  tlie  second  by  w,  and  add  the 

product  tu  the  tliird  e<juation,  we  shall  have 


{am  -f-  an  +  a")  x 
+  {bin  +  h')i  +  b")  //  \  =  //;»  +  Ini  +  7/'. 
4-  (6'?«  4-  c'n  +  c')  2; 


{!') 


In  order  that  tlio  quantities  y  and  2;  may  disappear  from 
this  ccpiation,  avc  must  have 

hn  +  Z/'w  +  h"  =  0, 
cm  +  c'n  -f-  c''  =  0. 

Since  wo  have  these  two  e([uations  between  the  quantities 
9U  and  n,  we  can  determine  their  values. 
Solving  the  equations,  we  find: 

b'c"  -  h"c' 


m  = 


n  = 


be'  -  b'c  ' 

b"c  -  be" 
Id  —  b'c ' 


These  are  the  required  values  of  the  multipliers.  Sul)sti- 
tutinjT  them  in  the  equation  {b),  we  find  that  the  coellicients 
of  //  and  z  vanish,  and  that  the  e({uation  becomes 


D 


,,  (//,."  -  b"e')  +  a:  {b"c  -  be")         , 


X 


•  _  hjb'c"  -b"c)  +  1,'{b"c-bc") 

-  y  -  b'c  +    • 


Clearing  of  denominators  and  dividing  by  the  coefficient  of 
X,  we  find 

li  {lie"  -  b"c)  +  ///  {b"c  -  be")  +  h"  {be'  -  b'e) 


X  ■= 


a  {b'c"  -  b'c')  +  a  {b"c  -  be")  +  a"  {be  -  b'c) 


EXERCISES. 


1.  Find  the  values  of  1/  and  z  by  the  above  process  for 


finding  x. 


d  71.       II'  WO 

md  luUl  tho 


(/O 


appear  from 


lie  (pumtilies 


iorf!.     Substi- 
ic  coelliciciits 


-  he") 


+  //'. 


c  coefficient  of 

be'  -yc) 


DYc  process  lot 


MULTJPLICATION  OF  SERIES. 


333 


For  this  purpnso  wo  mny  bopin  with  tlio  r(iuntion  {//)  and  find  vnliics 
of  m  and  n  such  tlint  tins  cocincionts  of  x  iiiid  2  in  (/O  shall  vuni.sli.  Tli«>so 
viiliU'H  will  l)c  (liHt-rt'iit  from  those  </\\i-n  in  (c'.  l?y  Hul)htitutiiii,'  ihi'iu  iu 
{b),  .r  und  2  will  be  fliiiiiuati'd,  and  we  filial!  «)i)tuiii  the  value  ol'  1/. 

We  then  find  a  third  set  of  values  of  m  and  n,  hucIi  that  the  coefli- 
cienta  of  x  and  y  shall  vanish,  and  thus  obtain  the  value  of  z. 

2.  Solve  by  the  method  cf  iiidetenniiuite  miiltiplierd  the 
exercise  3  of  §  140. 

MiiltlplicaHon  of  Two  Infinite  Series. 

284rt,  Proiji.km.  To  express  the  product  of  the  two 
series 


and 


The  method  is  simihir  to  that  by  wliicli  the  square  of  an 
entire  function  is  formed  (>5  17'-^,  2). 

We  readily  find  the  first  two  terms  of  the  product  to  be 

The  combinations  which  produce  terms  in  x^  arc 

Those  which  produce  terms  in  x^  arc 

In  general,  to  find  the  terms  in  a-"  we  befrin  by  multiplying 
ffo  into  the  term  Z'„.r"  of  the  lower  series,  and  tlien  multiplying 
each  succeeding  of  the  first  series  by  each  preceding  term  of 
the  second,  until  we  end  with  anb^x^.     Hence,  if  we  suppose 

Product  =  ^0  -f  A^x  4-  A^x^  +  .  .  .  .  +  AnX^  +  etc., 

we  shall  have,  for  all  values  of  n, 

An  =  a^bn  +  (tjbn-l  +  ^2^«-2  +  .  .  .  •    +  f/n^>o' 

Bv  giving n all intei^ral  values,  we  shall  form  as  manv  values 
as  we  clioose  of  An,  and  so  as  many  terms  as  we  choose  of  the 
series. 


334 


8EHIEH. 


!l-| 


i 


EXERCISES. 

1.  Form  llio  lu'odiict  of  the  two  scries: 

;/:'♦        ^        J^ 

Cfi         3fi  X^  , 

2.  Form  tlie  square  of  each  of  these  series. 

3.  Can  yon,  by  adding  the  sfjuares  tojTetlicr,  show  that  their 
sum  is  C(|ual  to  unity,  whatever  he  the  vahie  of  x't 

To  cfft'ct  tills,  multiply  onch  coeHicrieiit  of  X"  in  tlio  HUtn  of  tin;  wiuurcs 
by  11 1 ,  substitute  for  each  tenn  its  vulue  C'j'  giveu  in  g  257,  and  apply 
g'lJOa,  Th.  11. 

285.  Seriefi  procecdinr/  nccordinf/  to  tlic  Poicers  of  Two 
Variables.    Such  a  series  is  of  tiie  form 

do  +  ^o-'''  +  (f\y  +  CqX^  4-  h^:ty  -}-  a.y'^  +  etc., 

in  whicli  tiie  products  of  all  powers  of  x  and  //  are  comhitud. 
]$y  ci)llectin<5  the  coefficients  of  each  power  of  x,  the  scries  will 
become 

+  etc.,     etc.,      etc.,      etc. 

ITence,  the  series  is  one  proceeding  according'  to  the  powers 
of  one  variable,  in  "whicli  the  coellicients  are  themselves  series, 
proceeding  according  to  the  ascending  powers  of  another 
variable. 

Let  us  navo  the  identically  equal  scries  proceeding  accord- 
'  ing  to  the  ijsceuding  powers  of  the  same  variables, 

+  {B^  ^B,y-^B^y^  + )x 

■i-{Co  +  C,y+C,f  +  ....)x^ 
+  etc.,      etc.,      etc. 

Since  these  series  are  to  be  equal  for  all  values  of  x,  the 
coefficients  of  like  powers  of  x  must  be  equal.     Hence, 


BKllIKH. 


a35 


,v  thai  llicir 


157,  ami  uvi'b' 


'''o  4-  I'xH  +  I'^if  +  etc.  =  li^  -I-  //,//  -f.  /;^yj  I-  i-tc. 
etc.  etc. 

Af]faiii,  Hiiicc  tljosc  series  are  tu  be  e([iuil  lor  all  values  of  //, 
wo  imisL  have 

etc.  etc.  etc. 

Hence,  in  ojulcr  fJittt  tiro  srrirfi  pi'orrrdin'J  ffrrnrdin'J 
to  the  (tscciulin^  j)(>ircvs  of  tiro  luirlnhlrn  nttft/  l>r  idrnti- 
eitllij  vfjitdl ,  the  ro('/)}rientd  of  cvcrij  like  product  of  the 
powers  must  be  c(/uat. 


1 1 


||^ 


vers 


of  Two 


t'tC, 

Yve  {'omhiiu'il. 
the  series  \viU 


X 


t(i  the  powers 
liuselvcs  series, 
Irs  of  another 

ccaing  accord- 


1 


ialncs  ol  X,  the 
Hence, 


iV 


1  i  ! 


330 


SERIES. 


'» 


CHAPTER     I  I  I. 

SUMMATION      OF     SERIES. 


Of  Fijyurato  Xiimbcrs. 

280.    Tho  imiiihcrs  in  tliu  Ibllowing  ctjlunins  are  formed 
acconliiifi^  to  tliese  rules  : 

1.  The  (irst  column  is  composed  of  the  natural  luiiubcrs, 
1,  2,  3,  etc. 

2.  In  every  succeeding  column  each  number  is  the  sum  of 
all  tho  numbers  above  it  in  the  column  next  precedinf^. 

Thus,  in  the  second  column,  the  successive  numbers  are: 

1,     14-2  =  3,     1  +  2  +  3  :=  G,     1+2  +  3  +  4  =  lU,  etc. 
In  the  third  cohu m  we  have 

1,     1  +  3  =  A,     1  +  3  +  G  =  10,     etc. 

1 


1 

3 

1 

1 

3 

4 

1 

(J 

5 

1 

4 

10 

c 

10 

15 

7 

5 

15 

20 

35 

21 

G 

21 

35 

7 

etc. 

die. 

etc. 

(i) 


It  is  evident  from  the  mode  of  fornuition  that  each  number 
is  the  dilTerence  of  the  two  numbers  « 

next  above  and  below  it  in  the  col- 
umn next  foUowinjT. 

The  numbers  1,  3,  (I,  10,  elc.  i?i 
the  second  column  iire  called  trian- 
gular numbers,  because  they  repre- 


nl 


T\- 


fo 


N -  na+af4f5. 


SERIES. 


337 


re  formed 
uuiubei'S'* 


t\ie  sum 


of 


in<r. 


-  10,  etc. 


(-0 


,  ,u.h  number 


sont  miniliers  of  points  whieli  can  Lc  rogularly  arranged  u\i  v 
triangular  surfaces. 

Tlie  nunibor.s  1,  4,  10,  etc.,  in  the  third  columns  arc  callid 
pyramidal  numbers,  U'causc  each  one  is  composed  of  a  sum 
of  triangular  numl)er.s,  which  being  arranged  in  layers  over 
each  other,  will  form  a  triangular  jtyramid. 

All  the  numbers  of  the  scheme  are  called  figurate  num- 
bers. 

The  numbers  in  the  i'^*  column  arc  called  figurate  uumbers 
of  the  i'*  order. 

I^ST.  If  we  suj^pose  a  column  of  I's  to  the  left  of  the  first 
column,  and  take  each  line  of  numbers  from  left  to  right  in- 
clined upward,  we  .shall  have  the  successive  lines  1,1;  1,  2, 1 ; 
1,  ;},:>,  1,  etc.  These  numbers  are  formed  by  addition  in  tho 
same  way  as  the  binomial  coefficients  in  5$  171,  2.  We  may 
therefore  conclude  that  all  the  numbers  obtained  V)y  the  pre- 
ceding process  are  binomial  coelllcients,  or  combinat(^ry  expres- 
sions.   This  we  shall  now  prove. 


Theorem.     TIlc  h'^  number  in  the  i''*  culuinii  is  C(/ual 
toCT'~^ovto 

n{n-V  \){n  + '2) [n  +  /-!)_  ^x 

Proof.     Because  the  conil)inations  of  1  in  any  number  are 
eipial  to  that  numl)er,  we  have,  when  i  =  1, 

n^  number  in  1st  column  =  ii  =  ('", 

which  ajrrees  with  the  theorem. 

When  /  =  2,  wo   have,  by  the   law  of  formation  of   the 
numbers, 

7?'^  number  in  ^d  column  =  C'l  -f  C'l  +  T'l  +  .  .  .  .  +  Tj, 
which,  by  equaticm  {a)  (g  2(50,  3),  is  equal  to  C2'  • 

Therefore  the  successive  numbers  in  the  second  column, 
found  by  sup{)osing  h  =  1,  )>  =  2,  etc.,  are 

^i       fA       fA  f,n'1 


1 1  'J  t 


3t-4^: 


338 


SERIES. 


i 


*i 


:h 


Since  t!io  n^^  number  in  the  third  column  is  equal  to  the 
sum  of  all  above  it  in  the  second,  "\ve  have 

nff'  number  in  3d  column  =  ci-^(i-}-Ci-\-Cl'^  =  Cl'\ 

Avhicli  still  corresj)onds  to  the  theorem,  because,  jvhen  i  =.  3, 
n  -\-  i  —  I  =  n  -\-  2. 

To  prove  that  the  theorem  is  true  as  far  as  we  choos  to 
carry  it,  we  must  show  that  if  it  is  true  for  any  value  of  /,  it  is 
also  true  for  a  value  1  greater.  Let  us  then  sui)pose  that,  iu 
the  r^  column  the  first  n  numbers  are 


,ra 


,r-3 
r      > 


Since  the  w'^  number  in  the  next  column  is  the  sum  of 
these  numbers,  it  will  be  equal  to 

(  r  .  1  > 

which  is  the  expression  given  by  the  theorem  when  wo  suppose 
*  =  r  +  1. 

Now  we  have  ju-oved  the  theorem  true  when  /  =  3 ;  there- 
fore (supposing  r  =  3)  it  is  true  for  /  = -4.  Tiierelbre  (sup- 
l)osing  r  ■=.  4)  it  is  true  for  /  =  5,  and  so  on  indetinitdy. 

If  in  the  general  expression  (1)  we  put  i=.'i,  we  shall 
have  the  values  of  the  triangular  numbers  ;  by  putting  /  =  3, 
we  shall  iuive  the  [)yraniidal  numbers,  etc.    Therefore, 

mi  /A   i   •  1  1  ?'  ('^  +  ^) 

ihe  w"'  trianguhir  number  =  — -r    ,— ^• 
°  1-2 


The  n*^  pyramidal  number  = 


!il^i)  (^^  +  ^) 
1.2.3 


By  supposing  n  =  1,  2,  3,  4,  etc.,  in   succession,  we    find 
the  succession  of  triangular  numbers  to  be 


1.3      2-3      4-5 


1.2'     1.2'     1.2'     ""^''^ 

and  the  pyramidal  numbers, 

1^2-3      2.3.4      3.45 
i.2.3'     1.2-3'     1.2.3'     ^^'* 

which  we  readily  see  correspond  to  the  values  in  the  scheme  (A). 


SERIES. 


339 


al  to  tlie 


=    (  3      ♦ 


en 


1  =  3, 


dioos  to 
13  of  i,  it  is 
se  that,  iu 


the  sum 


of 


I  \SQ  suppose 

-=:  3 ;  thcre- 

lerclore  (^up- 

ttiuitc'ly. 

-  'I,  we  slv.iU 

ttiug  I*  =  3, 

I"*!  IV, 


o 

.sion,  Ave   find 


the  schome  (A)- 


Enunieratioii  of  Tritiiigular  Piles  of  Shot. 

288,  All  interostinn:  a])plic;iti(iii  of  tho  precodiuir  Hioory  is 
tlial.  ul"  Ihiding  tlu'  nuinbor  of  cannon-shot   in  a  pile.     Tiicrc 
jiiv  two  cases  in  which  a  pile  will  con- 
tain ii  ligurate  minibcr: 

I.  Elongated  projectile.^,  in  which 
each  rests  on  two  projectiles  below  it. 

II.  8i)herical  projectiles,  each  rest- 
ing oil  three  below  it,  and  the  whole 
I'urming  m  pyramid. 

Casi:  I.    Ehufjatcd  Project ilesi.    Here 
the  vertex  of  a  ])ile  of  one  vertical  layer  will  he  formed  of  one 
shot,  the  next  layer  below  of  two,  the  third  of  three,  etc. 
Hence  the  sum  of  w  layers  from  the  vertex  down  will  he  tho 
w'*  triangular  number. 

It  is  evident  that  the  number  of  shot  in  the  l)ottom  row  is 
Cfpial  to  the  number  of  rows.  Hence,  if  n  be  this  number, 
and  X  the  entire  number  of  shot  in  the  pile,  we  shall  have, 

n  {n  +  1) 

a 


N  = 


If  the  pile  is  incomplete,  in  consequence  of  all  the  layers 
above  a  certain  one  being  absent,  we  first  i'om[)tite  how  many 
there  would  be  if  tlie  ])ile  were  complete,  and  subtract  the 
number  in  that  part  of  the  pile  which  is  absent. 

Example.     The  bottom  layer  has  25  shot,  but  there  arc 

only  11  layers  in  all.     IIow  many  shot  are  there? 

05.20 
If  the  i)ile  were  complete,  the  number  would   be  —  ^  -• 

There  being  14  layers  wanting  from  the  top,  the  total  number 
of  shot  wanting  is    — ;, — •     Hence  the  number  in  the  i)ile  is 


N  = 


25. -^C  —  14.15        (14  -I-  11)  (IT)  -f-  11)  —  14.15 


a 


11(144-  15  +  11)  _^ 


220. 


r, 


340 


SElilES. 


Note.  TIiIh  piirtimlnr  prnMom  rnuM  Jiave  Tx^en  s<"»lve<l  more  briefly 
l)y  CDiisidprinfT  tlif  number  ol'  .shot  in  tlie  several  larers  a^^an  aritbnioticul 
progression,  but  wo  have  preferred  to  apply  a  ijeueral  method. 


f> 


'I 


EXERCISES. 

1.  A  jtilc  of  cylindrical  shot  1ms  n  iu  its  bottom  row,  and  r 
row.s.     How  many  shot  arc  there  ? 

2.  From  a  complete  i)ilc  having  h  layers,  «  layers  are  re- ' 
moved.     How  many  shot  are  left? 

3.  A  pile  has  n  shot  in  its  bottom  row,  and  m  in  its  top 
row.     How  many  rows  and  how  many  shot  arc  there? 

4.  A  pile  has  ;j  rows  and  h  shot  iu  its  top  row.  How  many 
fihot  are  there  ? 

5.  Explain  the  law  of  succession 
of  even  and  odd  numbers  in  the  se- 
ries of  triangular  numbers. 

6.  How  numy  balls  are  necessary 
to  fill  a  hexagon,  having  n  balls  iu 
each  side  ? 

Note.  In  the  adjoining  figure, 
n  =  'i. 

2S9.  Ca.se  II.  I'limmid  of  Balls.  If  a  course  of  balls 
Ik?  laid  upon  the  ground  so  as  to  fill  an  equilateral  triangle, 
having  n  balls  on  each  side,  a  second  course  can  be  laid  u})()n 
tlH'.<e  having  n  —  1  balls  on  each  side,  aud  so  on  until  we 
come  to  a  single  ball  at  the  vertex. 

Commencing  at  the  top,  the  first  courso  will  coui?ist  of  1 
ball,  the  next  of  3,  the  third  of  G,  aud  so  on  through  the  tri- 
angular numtjcrs.  Because  each  ]>yramidal  number  is  the 
.sum  of  all  the  preceding  triangular  numU'rs.  the  whole  num- 
ber of  balls  in  the  ?i  courses  will  be  the  /<**  pyramidal  number, 
or 

71  (n  4-  1)  Oi  4-  2) 


I 


W  = 


1.2-3 


EXERCISES. 

I.  How  many  balls  in  a  triangular  jwramid  having  9  balls 
on  each  side  ? 


■0  briefly 
hnH*ti<nil 


A',  and  r 
rs  arc  re- ' 
n  its  top 
.ovf  many 


•se  of  bulls 
triansl(?> 
laid  upon 
until  we 

onsist  of  1 

<r\\  the  tri- 

KT  is  the 

lolo  niuu- 

ul  uuniber, 


vinff  9  l^a^ls 


SERIES. 


341 


2.  If  from  a  triangular  pyramid  of  n  courses  k  courses  bo 
removed  from  the  toj),  how  many  balls  will  l)c  left? 

3.  How  many  balls  in  the  frustum  of  a  triangular  pyramid 
having  n  balls  on  each  side  of  the  base  and  m  on  each  t-idc  of 
the  upixr  course  ? 

Siiiii  of  the  Similar  Powers  of  an  Aritlmietical 

Proj^ressioii. 

290.   Put        «i,  the  first  term  of  the  progression; 
d,  the  common  dilference; 
n,  the  number  of  terms; 
m,  the  index  of  the  power. 

It  is  required  to  find  an  expression  for  the  sum, 

«T  +  (^'1  +  ^0"*  +  (^'1  +  -<''"'  + +  [f'l  +  ('i  -  1)  d^"", 

which  gum  we  call  Sm. 

Let  us  put,  for  brevity,  a^,  n„,  n^,  a^, ....  ff„  for  the  sev- 
eral terms  of  the  progression.     Then 

«2  =  (^  +  d, 

«3  =  «i  +  '^d  =  ffo  +  d,    • 

«R  ==  fit   +  {'i  —  l)d=  fln-l  +  d. 

Kaiiring  these  equations  to  the  {in-\-iy^  power,  and  adding 
the  c*iuation  (in^i  =  (hi  +  d,  wo  have 

(fm  I  -  am^t  +  {m  +  1)  a^d  +  i^^p-^  af-^P  +  etc. 


al 


m-l   —  rtfn^l 


=  ^.?^^  +  {>n  +  1)  (ifd  4- 


(i^  ^  =  <7^i  +  (w  +  l)a"'d  + 


a 


m  .1 


=  a 


w+l 


+  (m-f  1)W  + 


1-2 

{m 

-f-l)w 

1-2 

im 

4-l)/» 

1.^ 

« 

(m 

4-"l)m 

1-2 


^m-  1^/2  ^  etc. 


f(!^-hP  +  etc. 


^m-V/3  ^  (,tC. 


If  we  add  those  equations  together,  and  cancel  the  common 

tcnii«,    fi^^  +  o^'^  -\- -{-('lyK    which    appear   in   both 

niemlxTS,  we  shall  have 


342 


'I 


^\l 


SEJilES. 


c;i  --=  ^r'  +  0'^  + 1)^'^«  +  ^-^:p^'cPSm-i 


(w  +  l)m(m-l) 
H 1")') "  om-2,  etc. 


From  tliis  wo  obtain,  by  solving  Avitb  respect  to  /Sm, 


« 


7/1  f  1 


wi  (yy«  —  1) 


c            '*  ti        ^            ""7  0            '"  V"*  —  ^/   79  0  ^4^^     /n 

,Sm  ■=  -^— -r-^ —  dSjn-\ '   ^^      (/2,s^_3_etc.,   ( >) 

which  will  enable  us  to  find  S'm  when  we  know  ^S'j,  <S'„,  .  .  .  . 
xSm-i)  that  is,  to  find  the  sum  of  the  w'^  powers  when  we  know 
the  sum  of  all  the  lower  powers.  It  Avill  be  noted  that  *S'j 
means  the  sum  of  the  arithmetical  series  itself,  as  found  in 
IJook  VII,  Chap.  I  ;  and  that  *S'^  =  }i,  because  there  are  n 
terms  and  the  zero  power  of  each  is  1. 

By  §  209,  Prob.  V, 

To  find  the  sum  of  the  squares,  wc  put  m  =  2,  which  gives 


(3) 


291.  The  simplest  application  of  this  expression  is  given 
by  the  problem; 

To  find  the  sjiw  of  the  sqjim'cs  of  the  first  7i  natural 
numbers,  uanielij, 

I'M-  22  +  32  +  42  + +  ;?2. 

ITore  ^  =  1,  n„  =  n,  etc.,  S^  =  1  +  2 -\-}i  =  -^~ — -, 

so  that  (3)  gives 


_  («  + 1)3-1       n(?i-^l) 


71 

—  « 

3 


Noting  that  w  +  1  is  a  factor  of  the  second  member,  we 
may  reduce  this  e(juation  to 

_^;(>A  +  l)(2/?+l)  . 

which  is  the  required  expression  for  the  sum  of  the  squares  of 
the  first  n  numbers. 


SERIES. 


343 


-2> 


etc. 


>tc.,  [1) 

• »  5     •    •    •    • 

\'Q  know 
that  »S\ 
'ound  ill 
c  arc  ?i 


ich  gives 

is  given 
natural 


'2       ' 


mbcr,  we 
0) 


liiares 


of 


293.  To  find  the  sum  of  the  cubes  of  any  progression, 
we  put  m  =  3  in  the  equation  (•^),  which  then  gives 

Ad  2'''         '   ^        4: 


S,  =  -^.-j^  -  I  dS,  -  (PS,  -  ]  d\S,.  (0) 


Applying  this  as  before  to  the  case  in  whicli  a,,  a^,  a^, 
etc.,  are  the  natural  numbers,  1,  2,  3,  etc.,  we  tind 

{n-\-\Y  -  1       3  1 

'^3  =  -^ 2    -  "~  ^  ~  4    " 

—  (??  +  1)^  —  1  _  n  {n  +  1)  {:ln  +  1)  _  H^  ±V)  _  n 
_  _  .  ^  _        ^  ^. 

Separating  the  factor  w  +  1  and  then  reducing,  this  equa- 
tion becomes 


(5) 


But  — ^^- — -  is  the  sum  of  the  natural  numbers 


1  +  2  -f  3  +  etc., 

and  S^  being  the  sum  of  the  cubes,  wo  have  the  remarkable 
relation, 

13  +  23  +  33  4- +  w3  —  (1  ^  2  +  3  + 4-  n)\ 

That  is,  tha  smii  of  the  cubes  of  the  first  n  numbers  is 
equal  to  the  square  of  their  sudv. 

We  may  verify  this  relation  to  any  extent,  tliiis  ; 
When  /i  =  3,  P  +  23=:  1  +  8:^9  r=(l+2)^ 
When  ?i  =r  3,  13  4-  23  +  33  =  1  -!-  8  +  27  .-^  3G  rr  (1  +  2  +  3)'. 
When  «=4,  P  +  23  +  33  +  43^  1+8  +  27  +  04  =  100  =  (l  +  2  +  8  +  4)«. 
etc.  etc.  etc.  etc. 

29J5.  Eni(merafio)i  of  a  Rcctnmiular  Pile  of  JUiUs.  The 
prcc(^ding  theory  may  be  applied  to  the  enumeration  of  a  pile 
of  balls  of  which  the  base  is  rectangular  and  each  ball  rests  on 
four  balls  below  it.  Let  ns  put  p,  q,  the  number  of  balls  in 
two  adjacent  sides  of  the  base. 


''^^S^ 


.1 


344 


HEltlES. 


Tlien  the  second  course  will  liiivc  p  —  \  and  «7  —  1  halls 
on  its  sides;  the  third  p  —  'I  and  7  —  'I,  and  so  on  to  the  top, 
■\vhieii  will  consist  of  a  single  row  of  p  —  f/  -\-  I  balls  (sii})[)os- 
ihj,'  ])  ^  q).  The  bottom  course  will  contain  pr/  halls,  the  next 
course  (/>  —  1)  (7  —  1),  etc.  The  total  number  of  balls  in  the 
pile  will  bo 

To  find  the  sum  of  this  scries,  let  us  lirst  suppose  ]>  =  (/> 
and  the  base  therefore  a  S([uare.     We  shall  then  have 

JV'  =  q^  +  (-V  -   ly  +  {q  -  2)3  +  ....  +  1, 
which  is  the  svm  of  the  squares  of  the  first  q  numbers. 
Therefore,  :  v  ,  2.  ■    (4), 

■Hq -h  I)  {2q -\-  1) 


K' 


G 


(7) 


Next  let  us  pnt  r  for  the  number  by  which  p  exceeds  q  in 
the  general  ex})ressi(m  (0).     This  expression  will  then  hecome 

^=  7(7  +  0  +  (v-l)  (7-l  +  r)  +  (7-2)  {q-2  +  r)  +  .  .  .  . 

+  (1  +   •) 
=  fp  +  {q  -  1)''  +  il  -  2)2  +  ....+  2M-  1 

+  [7  +  (V  -  1)  +  (V  -  -'J  +  •  •  .  •  +  1]  r 


-  y('7  +  1)(2^  +  1)    ,    q{q+  1)  ^ 

7  (7  +  1)  jih'  +  ^y  +  1) 
G 


(§  ^'91,  4.) 


EXERCISES. 

1.  Find  the  sum  of  the  first  20  numbers,  1  +  2  +  3+  ...  . 
+  20,  then  the  sum  of  their  squares,  and  the  sum  of  their 
cubes,  by  successive  substitutions  in  the  general  cfpiation  (2). 

2.  Express  the  sum  and  the  sum  of  the  S(piares  of  the  first 
r  odd  numbers,  namely, 

1   +3   +5  +....  +  (2r-l), 
and  13  4.  33  ^  5'j  ^ _,_  (o,.  _  1  y^ 

3.  Express  the  sum  of  the  first  r  even  numbers  and  the 
sum  of  their  squares,  namely, 

2   +4  +  ()   + +   2;-, 

and  2=^  +  42  +  G^  + +  (2/-)2. 


i 


SERIES. 


345 


■  1  balls 
(he  top, 

[lu'  next 
Iri  in  the 


(7) 

cds  «y  in 
beeome 

'       ~f"     •      •      •      u 

■  (L  +   •) 
.  +1]^' 


t)  "J"    •    •    •    • 

of  their 

ion  {'I). 

'  the  iirst 


5  tmd  the 


4.  A  rcctanj^nlar  pile  of  hulls  is  started  with  a  ha«o  of  p 
halls  on  one  side  aiul  7  on  tlie  other.  How  many  halls  will 
IhtTe  he  iu  the  pile  after  3  courses  have  been  laid  ?  How 
many  after  .s  courses  ? 

5.  Find  the  value  of  the  expression 

3.-5 
1  {a  +  Ox  +  cx'i). 

X-l 

6.  Find  the  value  of 

X=:f> 

1  {a  +  Ijx  -f  cx^). 

x=l 

29-1.  To  find  the  sum  of  n  terms  of  the  scries 

1.2^2.3^3.4^ ^n{HA-\) 

Each  term  of  this  series  may  be  divide,  ii;  j  two  parts, 
thus : 

J__l_]  jL_i_l 

1-2  ~"  1       2'  2-3  ~"  2       3' 

1__  _  1 1 

n  {n  +  1)  ~  n      71  -\-  s. 

Therefore  the  sum  of  the  scries  is 

in  which  the  second  part  of  every  term  except  the  last  is  can- 
celled by  the  first  jiart  of  the  term  next  following.  Therefore 
the  sum  of  the  n  terms  is 

_      1 _      w 

If  we  suppose  the  number  of  terms  n  to  increase  without 

lin)it,  the  fraction r  Avill  reduce  to  zero,  and  we  shall  have 

n  -{-  I 

T~o  +  T'-i  +  •!   (  +  ^^^-y  ^'^  litjudtum  —  1. 

This  is  the  same  us  the  sum  of  the  geometrical  progression,  „  +  ^  +  q 

«       4       9 


^>Jis»'    ■»ar«wr'"-"jn  iiw.w#wl 


M 


m 


•I 


i  j  H' 


340 


SEIUES. 


+  rtc.,  «'/  infinitum.     It  will  l»o  iiitcrrstiiip  to  com  pure  the  firnt  few  tiTina 
ol  the  two  Bcrius.     'J'lu'y  art! 


1       1 


1 


1 


2  ^  0  "^  lii  ■*■  20  ■*■  yo  '   4^ 


1         1 


1       1 
4 


1 

H 


1 
1(5 


4-  -f-  -J.  4-  J.  . 


1 


I 

U4 


Wo  8P0  that  the  first  tcriii  is  the  sanie  in  l)oth,  wliih*  tho  next  throo 
lire  laiT^cr  in  the  ^n-diiiftiical  pro^^n'ssion.  At'ttT  tin-  ftnirtli  tcriu,  tliu 
tt'mi.s  of  tlu'  i»rot^ixvsaiua  l)ecomc  the  HinalliT,  and  coutinuu  ho. 

tiJ)5.  (Ivticridizdlion  of  the  Preccdim)  UcsuU.    Let  us  take 
the  scries  of  whicli  the  w'^  term  is 

P 


V 


{i  +  n-\){j  +  n-\) 
The  scries  to  n  terms  will  then  bo 
P  .  P 


{i  +  1)  0'  +  1)  +  {i  +  2)  (7  +  ^>) 

+ 


"J"     •    •    •    • 


P 


{i  +  w  -  1)  {j  +  H-1) 
If  WG  supi^osc  j  >  i,  and  put,  for  brevity, 

^  =  j  —  h 
the  terms  may  be  put  into  the  form 

p^pn_n 


P -Pi    ^      _     M 

(/+  1)  ~  lAi  +  I       /+  1/' 


ete. 


+  I       7  + 
etc. 


(/  +  n 


_v ^pi    i i__.y 

1)  (./  +  'i  4-  1)       ^"  V'"  ■\-  n  —  V      j  +  n—  1/ 


When  we  add  these  quantities,  tlie  second  part  of  each  term 
Avill  be  cancelled  by  the  tirst  part  of  the  y^  term  next  follow- 
ing, leaving  only  tho  first  part  of  the  first  k  terms  and  the 
second  part  of  the  last  k  terms.     Hence  I  he  sum  will  be 


k\i  ^  ■• 


i+i ' 


+ 


1 


1 


J  +  l      i-\-n      i  +  n—1 


;  + 


n-l) 


4 


f^EIilES. 


347 


Example.    To  find  the  sum  of  n  torms  of  tlie  scries 

±,±^±,±.         .    JL__ 

2-5  ■^.•J.(}  "^4.7  "^5.8"^ "^(n4.i)(;i  +  4)* 

Kuch  term  may  be  expressed  in  tlie  form 

3-0      a\;i     (;/' 

4. 7  ~  3  \4       7/' 

-__L-_  =  1(1 L) 

n  {h  +  3)        3  \;i      n  +  3/' 

J_____^VJ 1   \ 

{n  +  1)  {n  +  4)        3  \n  +  1       w  -f  4/ 

Tlioroforc,  scparatin<}f  the  positive  and  negative  terms,  wo 
lind  the  sum  of  the  series  to  be 

1/1111  1  1 

3^  "^  3  +  4  "^  5  ■^••"  +  w  +  ^TTl 

_1_1_      _1 1 1 1^ 

5       G      ""      71      71  +  1      n-i-'i      n-\-'3      w  + 

or,  omitting  the  terms  whicli  cancel  each  other, 

3 \2  "^  3  "^  4       n  4-2       7i  +  :i       nA-\l' 


u 


+  a       n  + 
"When  n  is  infinite,  tiie  sum  becomes 

3\-^  ^3  ^  4/        3  i'i       30 


EXERCISES. 

What  is  the  sum  of  n  terms  of  the  series 
1  1 


1_ 
1 


0'4       4'5       5-0 


2. 


_L        J_ 

•      c     ry    "T    ^Tk    T"    •   •   •   •    "T 


3-5   ■   5-7   '   7-9 


(3w  +  1)  {In  -f  3) 


l!>. 


;  ^^^sm^^^ 


i4i^ii-umf^B^^p^^^"wwww« 


il 


»» 


'MS 


ISEIUKS. 


-•J       •{•<>       4-7  ^  (//  4-  1)  (,i  4-  4) 


:i 


3 


3 


5-  Slim  (lie  scries 
1  .  1 


+ 


-.n  + 


-rrr  4-  etc.,  ff^/  inf. 


a(a  +  1)  ^  (a  +  l){a  4-  ^1  "^  («  +  ^)  (^<  +  3) 
Jil>(>.    To  sum  tlie  scries 

,S'  =  1  +  2r  +  3ra  +  4/'3  +  etc. 

Let  lis  first  fiiul  the  sum  of  n  ivnm,  which  wo  shall  cull 
/S'n.     Then 

Sn  =  l-\-2r  +  3r2  +  4y3  + nr'^-K 

;Miiltiplyiii<r  by  r,  wc  have 

rS\  =  7-4-  2/-2  +  3/-3  4-  4/"'  4- 4-  m"^. 

By  sul)tractioii, 

(1  —  r)  Sn  =  1  +  r  4-  7-2  4-  7-« 4-  r«-i  —  7ir>' 

1  —  r'» 


Therefore,       iS>i  = 


1-r 
1  —  ?•» 


«r«  (§  2V^,  Prob.  V). 


(1  _  ry      1-r 

Now  Pupposo  n  to  increase  without  limit.  If  r^  1,  (ho 
sum  of  the  series  will  evidently  increase  without  limit. 

If  }'  <  1,  ])oth  /•«  anil  ;?/•"  will  converge  toward  zero  as  n 
increases  (as  we  shall  show  lu>reaffer),  and  we  shall  have 

1 


S  = 


{\-ry 


EXERCISES. 

Find  in  the  above  way  the  sum  of  the  following  series  to  n 
terms  and  (o  inlinity,  supposin^j  /•  <  I  : 

1.  rt  4-  '3((r  4-  5((r^  4-  r<//-a 4-  (O;^  _  1)  ar»~K 

2.  2a  4-  4ar  4-  6a/^  4-  Sai'^.  ...  4-  ',l)/ar''-\ 

3-     («  +  b)  r  +  {a  4-  2b)  i^  + -\-  {n  +  7ib)  r«. 


1 


sh'itim 


31!) 


v-^  to  n 


4 


!il)7.    »Surn  the  series 


+  irrn  +  rrr-K  +  etc., 


1.2.3  '  2-3.4  ■  3.4.5 

1 


(") 


of  ubicli  tlie  L'eiKTal  term  is  — -. , ,  ,        -  ,^- 
/<  (w -|- 1)  (/^  +  «) 

Let  us  find  whether  we  euii  cx[>iv>s  this  series  jis  the  sum 
of  two  scries.    Assume 

1 A__      B^ 

n  (n  +  1)  (w  +  2)        n  {n  -}-  1)  "^  {n  +  I)  [n  -f  2)  * 

Avhero,  if  possihle,  the  vuhies  of  the  indeterminate  cocnicicnts 

A  and  //are  to  be  so  chosen  that  this  e<[nution  shall  be  true 

ident  icily. 

liedueing  the  second  member  to  u  common  denominator, 

we  have 

1 _  (.4  -f  /?)  n  +  2  J  ^ 

71  {n  -\-  I)  {)i  4-  2)  ~  n  {11  -f-  1)  (n  +  2)' 

In  order  that  these  fractions  may  be  identically  equal,  we 

must  have 

{A  +  B)n  -{-  2/1  =  1,  idcitlicalhj, 

which  rcHjuires  that  we  have  (§  281), 

^  +  ^  =  0,        2.1  =  1. 


This  gives 

Therefore, 
1 


o» 


^  =  -i" 


n  {n  +  1)  {n  4-  2)        2  n  {n  +  1)       2  {n  +  I)  {n  +  2)  * 

so  that  each  term  of  the  series  (a)  may  be  divided  ixito  two 
terms.     The  whole  series  will  then  be 

I  /  1  1  1  .    \       1  /  1  J  I  ,    \ 

2(lT2  +  2:3  +  3^  +  '^''1  -  2I2.3  +  3:4  +  4:5  +  '''V' 

Wo  see  on  sight,  that  by  cancelling  equal  terms,  the  sum  of 


n  terms  is 


>S'n  = 


1 


1 


4      2(/iH- l)(/i  +  2)' 
and  the  sum  to  infinity  is   .• 


3.50 


SERIES. 


2J)8.  CoiLsidor  tlio  liiinnoiiic  scries 

111. 
1  +  ,  +  ,  f  ,  +  etc., 

of  which  tho  «'*  term  is  •  This  series  is  iliverfjoiit,  hecuusc 
we  iiKiy  divide  il  into  au  uiiHiiiitcd  munber  oi"  j>urts,  each 
eciiial  to  or  greater  than    j.  as  Ibiluws: 

1st  term  =  1,     >  ^  ; 


r 


2d  term  = 


r 


3d  luid  4th  terms  >     ; 


etc. 


etc. 


Ill  general,  if  we  consider  the  n  consecutive  terms, 
_1_  1  1 

„  +  1  +  ,r+  ;j  +  •  •  •  •  ^-  ^lui 
1 


{a) 


the  smallest  will  be  .-- ,  uad  therefore  their  sum  will  he  greater 
than    /     X  n.  that  is,  greater  than  -• 

v/l  /v 

N«tw  if  in  {(t)  we  suppose  n  to  take  the  successive  values, 
1,  •.',  I,  s,  Ift,  etc.,  wc  shall  divitle  the  series  into  an  unlimiti'd 

numher  of  parts  of  the  form  (n),  each  greater  than   -•     There- 
fore, the  sum  has  no  limit  and  so  is  divergent. 


'I 


■  \ 


Ol"  DinV'rciK'cs. 

*'1M).  Wlh-n  we  have  a  series  of  (|iuintities  proceeding  ac- 
cording to  any  law.  wc  m;iy  lake  the  liilj'crence  of  c\ery  two 
consecutive  (pumt'lics,  and  thus  form  a  series  of  dillereiiees. 
The  terms  of  this  seriv's  are  called  First  Differences. 

'i'aking  the  dilTcrence  of  every  two  cmisecutive  dilTcrences, 
we  shall  have  another  series,  the  terms  of  which  are  called 
Second  Differences. 

The  proce.-..>  may  be  continued  so  long  as  there  are  any  dif- 
ferences to  write. 


,  lu'cause 
rts,  cacli 


0  greater 


0  values, 
nliiiiitoil 

There- 


'(liiijj;  ac- 
vcry  (wo 
li'iviiccri. 

oronces, 
ii-  callt'd 

any  dif- 


BEltIP:S. 


351 


ExAMPi.i:.     In  the  secoml  column  of  the  f(^llowiug  tal)ie 
are  given  the  seven  values  of  the  expression 

for  ./•  =  0,  1 ,  iy  3,  4,  .J,  'I. 

\n  the  third  col"   'in  a'  are  given  the  diirereiices, 

C  —  ib  =  —  11),      ]_(;-—.>,     —  14  —  1  =  —  l'>,    etc. 

In  column  a"  are  given  the  diireivnees  of  these  dillerenees, 
nanielv. 


_5-(-l:.)  =  +  14, 


15 -(-o)  = 


X 

0 
1 

8 
4 

5 

6 


+  :^a 
+  C 
+    1 

—  14 

—  39 

—  50 
+    1 


—  10 

—  5 

—  15 

—  25 

—  11 
+  51 


+  U 

—  10 

-  10 
+  14 
+  (J2 


A'" 

—  'M 
0 

-f  48 


A'* 


10,     etc. 


+  v>4 

4-24 
-f  ri4 


0 
0 


The  process  is  continued  to  the  fotirth  order  of  di (Terences, 
which  are  all  eijiiah  whiiiee  those  of  tiie  lil'th  and  lolloNsing 
unk-rs  are  all  zero. 

It  will  \>^.  noted  that  the  sign  of  each  dilTi'rence  is  taken  so 
(hat  it  shal'  expivss  each  <|uantity //////^^^•  the  (|uaii(ity  next 
preceding.     We  have  tlu-rel'orc  the  following  delinitions  : 

:{0().  Drf.  The  First  J^ifference  <d'  a  riuictioii  of 
any  variable  is  the  iiicn'iiiciit  of  the  I'mictioii  catiscd  by 
an  incnMucnt  of  unity  in  the  variable. 

The  Second  Difference  is  the  ditferenco  between 
two  cunseciiti\  e  jlrst  d ill* 'fences. 

In  pMieral.  tin'  >*"'  Difference  is  the  diflereiice  be- 
tween two  consecutive  [^n  —  1  r'  dillerenees. 


352 


SERIES. 


To  invostifj^atc  the  relation  amoiif]^  the  dilTeronces,  let  us 
represent  tlie  successive  nnmher.s  in  each  column  l)y  the  indices 
1,  -Z,  o,  etc.,  and  let  us  jiut  Aj,  Ao,  A3,  etc.,  for  tlie  values  of 
0r.  We  shall  then  iiave  the  following  scheme  of  ditlerences, 
in  which 

a;=a,  —  Ao,  a;=:Ao-a,,  a;  =  a3-a2; 
a^  =  a;-a;,  a'; -a; -a;,  a;  =  a;-a;; 
a;  =  a;-a;,   a';=a:-a';,   a::=a;;-a;; 

etc.  etc.  etc. 

tlie  w'*  order  of  differences  being  represented  by  the  symbol  A 
with  n  accents. 


"1 
As 
A, 


a: 


A', 


^: 


I 


An-l 


•I 


A„ 


Let  ns  now  consider  the  following  problem : 

To  express  At  in  terms  of  Aq,  Aqj  Aq,  etc. 
We  have,  by  the  mode  of  forming  the  dillerences, 
A,   =  A„  -f  a'u,     a',   =  a'o  -(-  a'^,     a','  =  A'o  -}-  a';,  etc.       («) 

Aj  =  A,  H-  A,,     As  =  Ai  +  A,,     As  =  A,  +  A      etc. 

If  ill  this  last  system  of  e(|Uations,  we  substitute  the  values 
of  A,.  A,,  etc.,  from  the  system  {a),  we  have 

Ao  =  A„  +  -^a;  -f  a'u,     Ag  =  a'u    I    -iAu  -b  a';,  etc.    {!>) 

Again, 
Aj  =  A^  +  A,.,     A3  =  a',  -b  ^'i,     A3  =  Ag  +  A3,  etc. 


DIFFERENCES. 


353 


?s,  let  us 
c  indices 
allies  of 
Icrences, 

ymbol  A 


Ic.      {a) 

tc. 

le  values 

etc.    (/>) 

^g,  etc. 


Substituting  the  values  of  Ao,  Ao,  etc.,  from  {h),  wc  have 


A3  =  Ao  +  --iAo  + 


4-  a'o  4-  JiA;  +  a'; 


or 


A3  :=  A„  +  ;?a;  +  ;ja;  +  a; 

A'3  =  a;  +  2^',  -t-  A'; 
f  ^0  +  2a;'+a'j 


(# 


a;  =  a'o  +  3a;  +  :u;'  +  a-; 

Forming  A^  =  A3  +  A3,  etc.,  we  sec  that  the  coefficients 
of  Aq,  a'o,  etc.,  which  wc  add,  are  the  same  as  the  coellicients 
of  t!ic  successive  ])i)wers  of  x  in  raising  1  4-  x  to  the  n*'^  jxjwer 
by  successive  multijilication,  as  in  jj  i;i.  That  is,  to  form  A^, 
A'^,  etc.,  the  coellicieiits  to  Ijc  added  are 

1     3     .3     1 
1_  3_  3_1 

14     0     4     1 

and  these  arc  to  be  added  in  the  same  way  to  form  A5,  and  so 
on  indrlinitelv.  Hence  we  conclude  that  if  i  be  any  index,  the 
law  will  be  the  same  as  in  the  binomial  theorem,  namely, 


Ai  =  Ao  +  /a'o  +  (^)  a';  +  (I.)  a'o'  +  etc. 
Ai  =  A'o  4  iK  +  (I)  ^0'  +  (;j)  ^0  +  ^'tc 


('0 


To  sho'/  rigorously  that  tliis  result  is  true  for  all  values  of 
/,  we  have  to  prove  thut  if  true  for  any  one  value,  it  must  be 
true  for  a  v  !ue  one  greater.  Now  we  have,  by  delinition, 
whatever  be  e, 

Ai,i  =  Ai  4  Ai',         aI-i  =  aI  +  a!',      etc. 
Jlenee,  substituting  tlu'  abovi"  value  of  A*  and  M, 


Ai.i  =  Ao    I    (/  4  !)  a'o  +  I^Q  4  /Ja'; 


(«) 


23 


354 


SERIES. 


M 


Wc  readily  prove  that 

Q + (;) 


en 


etc. 


etc. 


Suhf^tituting  these  values  in  (f),  the  re>nlt  is  the  same  given 
bv  the  r(|iiation  {(f)  when  we  put  /  -f  1  for  i. 

The  forni  (c)  hIiows  the  lorniula  to  l>e  true  for  /  =  3. 

Therefure  it  is  true  for  i  —  4. 

Th.-refure  it  is  true  fur  i  =  5,  etc.,  intlefiniiclv. 

EXAMPLES    AND     EXERCISES. 

1.  Having  given  1^  =  7,  a;  =  5,  YJ  =.-  -  2,  and  A",  A'', 
etc.  =0,  it  is  required  to  find  the  values  of  A,.  A..  A,,  etc., 
indefinitely,  both  bydireetcuniputatiouaud  by  the  fonnula  (</).' 

We  start  the  work  thus: 


TIk-  numbors  in  coluiiin  A"  arcnfl 
equn!  to  -  2,  ht'causo  A'"  -  0. 

Each  iiuinl)er  in  column  A'  nftor 
tlip  first  is  found  by  adding  A"  or  —  )i 
to  tlu-  one  next  above  it. 

Each  value  ot  Ai  is  tlien  obtained 
from  the  one  next  above  it  by  uddin" 
thi-  appropriate  value  of  A^. 

This  proctps  of  addition  ran  be 
carried  to  any  extent.  Continuin-^-  it 
to  I  =  10,  we  shall  linil  A^  =  — ou 


1 

0 


3 
4 

i-te. 


+  12 
+  15 
etc 


+  5 
+  3 
+  1 
—  1 
etc. 


2 

2 


etc. 


Xext,  tlie  general  frniiih.  (rh  ;,-  ,cj^,  hy  putting  A^  :=  7, 
^'o  =  ^»  ^'i  =  —  2,  and  all  following  values  =  0, 

A,  =  7  +  5/-2ii^^:^ 

and  the  .<tudent  is  now  lo  siiow  that  by  putting  i  =  1,  !  =  2, 
etc.,  in  this  expression,  we  ol>tain  thc'same  values  of  a,.  Ag! 
As,  .  . . .  Aio,  that  we  get  In  addition  in  tlie  alxive  scheme. 

It  is  moreover  to  he  remarked   that  wt  con  reduce  the  last 
equation  to  an  entire  function  of  /,  thus : 

Ai  =  7  +  Gi  -  i\ 


IJIFFERENCES. 


35i; 


7.    Having    given     Aq  =  5,    A'^  r=  —20, 


^;;  = 


nc  given 


3. 


A'",  A", 
A;,,  ete., 

lUlil  {il). 


^i 


:5'», 

j.'^'  =r  +  !♦,  it  is  required  to  find  in  the  same  w  ly  flic  values 
of  A,  to  ^5,  and  t'^  express  Ai  iio  uu  Milire  I'unciion  *ii'  i  by 
fumiiila  (//). 

3.  On  Mareli  1,  IS.Sl,  at  (ireenwieii  noon,  the  sun's  lon^'-i- 
tnde  was  34r  5  lO'.lJ  ;  on  March  'i,  it  was  greater  l»y  1  0  U  '.C, 
but  this  dailv  increase  was  diminishin*'  !)>•  'Z"  each  dav.  It  is 
re<|uired  to  comitute  the  longitude  lor  the  first  seven  nays  ot* 
the  month,  and  to  tind  an  expression  for  its  value  on  the  n^'^ 
day  of  March. 

4.  A  family  had  a  reservoir  eontaining,  on  the  morning  of 
3fay  .5,  V3')  gallons  of  water,  to  which  the  city  a.i'k'd  regularly 
5f)  gallons  per  day.  IMie  family  used  3.")  gallons  on  May  5, 
ami  5  gallons  more  eaeli  sulisecjuent  day  than  it  did  on  the  day 
juvcT-ding.  Find  a  general  expression  for  the  (plant ity  of 
uater  ou  the  //''  day  of  May;  and  hy  tMnititinir  this  expression 
to  Zen*,  find  at  what  time  the  water  will  all  be  gone.  AUu  ex- 
plain the  two  answers  given  by  the  e(puition. 


A. 


etc. 


k         IV 


I  =  '4, 

Aj .  Ag* 

ne. 

be    last 


Tliooi'CMiis  of  niffcroiicoH, 

.*501.  To  investigate  the  general  properties  of  differences, 
we  ujsC  a  notation  slightly  diHerent  from  that  just  employed. 
If  u  )Mi  any  function  of  x,  which  we  may  call  <pj;  so  that 

we  put 

u  =  (px, 

then  AW  =:  0  (.r  -f-  1 )  —  (px.  {(() 

Hero  the  pymbol  A  does  not  re])resejit  a  multiplier,  but 
merely  the  words  (fijft'roirc  of. 

The  second  difference  of  u  being  the  difference  of  the  dil- 

fen-nee.  may  l>e  represented  l)y  AA/^ 

For  brevity,  we  i)ut 

A'',*^  for  llii, 

wherf*  the  index  ti  is  not  an  exponent,  but  a  symbol  indicating 
a  ^eeond  difft-n-nce. 

Continuing  the  same  notation,  the  vt''*  difference  will  be 
rc'pre«>enied  by  A''. 


350 


SERIES. 


K  X  A  M  P  L  E . 

To  find  the  sncccssivo  dilFercnces  of  the  function 

n  =  (tx^  +  bx^. 
By  the  formula  (a),  we  have 

Ml  =  a  {.c  +  1)3  +  ./  {x  -f-  1)3  _  rta:3  _  j^^.2 . 
ami,  by  developing, 

AH  =  Sax'i  +  (3rt  +  2b)  x -\-  a  +  b. 
Taking  the  difference  of  this  last  ecpuition, 

A'';«  =  'da  {x  +  1)2  +  (3rt  +  2b)  {x  +  \)  +  a  +  b 

—  3rt.6«  —  (;3rt  +  2b)  x—a  —  b 

=  Grt.c  +  r.a  +  ^6. 

Again  taking  the  dilferencc,  we  have 

A''/<  =  ()(/  {x  +  1 )  —  O^f.T:  =  Grt. 
This  expression  not  containing  x,  A%,  A''//,  etc.,  idl  vanish. 

EXERCISES. 

Compute  the  differences  of  the  functions  : 

I.     7^  ^-  mx^  +  nx  +  p.  2.     2x^  +  3.1-2  ^  5, 

3.  r)./3  _^  io,i'.'  ^-  15. 

4.  In  the  case  of  the  last  expression,  prove  the  agreement 
of  results  by  computing  the  values  tsf  Au,  A^ii,  etc.,  for  x  =  0, 
r=:  1,  and  e  =  3,  and  comparing  them  wiih  (hose  obtained 
by  the  method  of  §  r>'.)i).  The  latter  arc  shown  in  the  follow- 
ing table: 


a 


u  =z  5.r3  -f-  lo.t2  4.  15. 


A^H 


id 

t 

30 

15 

50 

3 

m 

«S 

SO 

30 

3 

uo 

U5 

110 

30 

4 

4'»5 

255 

5 

ii 


nrFFERbJNCES. 


357 


a-h 


vanish. 


jrccmont 
r  a:  =  0, 
ohiaiiKMl 
c  foUuw- 


5.  Do  tlio  saTno  thing  for  exercise  2,  and  lor  the  runeiioii 
tuhiilatecl  in  §  :3U'J. 

IMVl*  It  will  1)c  seen  by  the  preceding  examples  and  exer- 
cises, that  for  I'aeh  ditTerence  of  an  entire  function  of  ./•  wliicli 
■vvo  form,  the  degree  of  tliu  function  is  diniinisjied  by  unity. 
This  result  is  generalized  in  the  following  the-oreni: 

Tlie  n^  di/fercnccs  of  tlie  J'luictloii  a"  are  constant 
and  i'f/ioal  to  n\ 

Proof.  If  u  =  a'S  we  have,  by  the  dctinition  of  the  sym- 
bol A, 

A?«  =  (.r  +  1)"  —  .T», 


or 


Au 


=  ;u"  1  4-  (j.<"~^  4-  etc. 


That  ii^:,  iti  taK'lng  the  (lij)'evcnce,  the  hi'Jhest  poii'rr  of 
X  is  uialti plied  by  its  eA'punent  amt  the  tatter  is  dimin- 
ished Ijij  unitij. 

Continuing  the  process,  we  shall  lind  the  n^''-  dill'erenco 

to  be 

n  (n  —  1)  {n.  —  2) I  z=  nl 

Cor.    If  we  have  an  entire  function  of  x  of  the  degree  w, 

az"'  +  tjx>'-^  +  ('./•"-■'  +  etc. . 

the  {h  —  1)"'  diirerencc  of  />.r«-->,  the  {11  —  iiy^  dift'tTenco  of 
(;j;n-i^  etc.,  will  all  be  constant,  and  therefore  the  n^^'-  ditlVrenco 
of  these  terms  will  all  vanish.  'J"  here  lore,  the  n"'  dilference  of 
the  entire  function  will  lie  the  same  as  the  ;i'''  dilference  of 
ax"' ;  that  is,  we  have 

A"  (ra-"  +  ix"-!  4-  etc.)  =  an  ! 

Hence,  tJie  n^'*-  differenee  of  a  fanetion  of  the  n^i*'  de- 
gree is  eonstaut,  and  (upud  to  n\  wutli filied  btj  the  coeffi- 
cient of  the  higlicsb  power  of  the  uariable. 


i 


358 


SEUIEti. 


'\ 


CHAPTER    IV. 
THE     DOCTRINE     OF     LIMITS. 

JJO.'J.  Tlu' doc'triiic  of  limits  ciiiltraros  ji  set  of  i)rin('ii)los 
upplicablo  to  cases  in  which  the  usual  nicthocls  of  calculaiiou 
fail,  in  ('onse(iucucc  of  some  of  the  (quantities  to  be  useil  van- 
ishing or  increasing  without  limit. 

Wc  luive  already  made  extensive  use  of  some  of  the  ])rinei- 
ples  of  this  doctrine,  and  thus  familiarized  the  student  with 
their  a])j)lication,  hut  our  fni'ther  advance  retiuires  that  they 
should  be  rigorously  developed. 

A\ro:\r  I.  Any  qtitiiitity,  liowcvcr  smnll,  iiiiiy  l)o 
multiplied  so  ol'tcn  as  to  cxcwd  any  otlicr  lixcd  cjiian- 
tity,  li()\v(n(T  ,<i:n'at. 

A\.  11.  iJonrcrsi'Ji/,  any  ([iiaiitity,  Itowcvor  or(.;it, 
may  be  dividt'd  into  so  many  ]).»rts  that  cacli  part  shall 
l)L'  less  than  any  other  fixed  (piai:tity,  however  small. 

]>('/.  An  Independent  Variable  is  a  (piaiility  to 
wliicli  we  may  assign  any  value  wo  j)lease,  however 
small  or  great. 

'rni':oiii:M  \.  If  (i  fraction  Inti-c  (iiifi  finite  vuino'u tor, 
and  an  independent  rariable  for  its  (tenoniinatttr,  ire 
ma'i  (tssi'Jn,  to  flu's  denoiiiinator  a  T<due  so  ijreitt  tlnit 
^he  traction  shall  he  less  than  any  (/aantiti/,  howerer 
small,  a'hicli  ive  inaij  assign. 

Proof.  Let  a  he  tlie  numerator  of  ihe  IVacI ion.  .r  its  de- 
nominator, and  a  cany  ([uantity  however  small,  which  we  may 
choose  to  assign. 

Let  li  he  the  num])er  of  times  Ave  must  innlti])ly  a  to  make 
it  greater  than  a.     (Axiom  I.)     We  shall  then  have 

(t  <  Uh, 

Conscfpiently,  -  <  «. 


I, 


I 


I'iiicipU'.s 
c-iilaiioii 

St'll  Vllll- 

e  priiici- 
L'lit  with 
lat  tlicy 

may  ho 
.1  quaii- 

r  <iT('.'it, 

irt  sliiill 

iiuill. 

utlt}'  to 
I  ()  wo  vol' 


irrttfor, 
for.  irf 
1/  that 
oiL'ci'cr 

r  its  (lo- 
wc  iiuiy 

o  miikc 


\ 


LIMITS. 

Ilcncc,  by  taking  x  greater  than  w,  wc  shall  have 

a 


350 


X 


<  «. 


Example.     Lcl  a  =  10.     Thtii  il"  wo  take  Tor  «  in  f'ucees- 


Bion, 


I 


I 


I 


loo'    ii>,()UU'    1,(H)(),()(I() 


,  etc.,  we  havi'  onlv  to  lake 


X  >  1,000,     X  >  100,000,     X  >  10,000,000,     etc., 
to  make     -  less  than  tc. 

X 

Tn  the  lan^^nage  of  limits,  t he  above  theorem  is  expressed 
thus  : 

21)c  limit  of  ,  when  x  is  indrfnitclij  inrrcftscd,  is 
zero. 

Tin'oUFM  II.  //'ti  frnclion  hdi'c  ((inj pnitr  niinirrfffov, 
and  (in  inili'/x'ndmf  rrtriah/f  for  its  drnnnii nahir,  wn 
mnii  iissi'Jii  Id  lliis  dcninniiiftfur  ft  mini'  sd  snndl  Unit, 
iiw  I'nirtinn  sliiill  c.vci'cd  any  (/aantiti/,  /loircrcr  ^jrcat, 
which  awniay  assii^n. 

Proof.     \\\{  as  Ixforu       for  llu,'  IVaftion,  and  k't  A  bo  any 

liumbor  however  great,  which  we  <'1ioose  to  a-sign. 

Let  n  be  a  number  greater  tlian  A.  Divide  a  into  ti  parts, 
and  let  «  bo  one  of  these  parts  ;  then 

a  =  im. 


Couseqm'ntly, 


a 


=  n. 


Therefore,  if  we  take  for:«  a  (piantity  less  than  a,  we  shall 

have 

a 

X 

a 
x 


or 


>  n  >  J, 

>  A. 


"Ri'M.     Tf  we  have  two  independent  variables,  .r  and  y: 
\\\  may  make  ./•  any  numbir  of  times  greater  than  >/. 


300 


LfMim 


'I 


TlxM  wo  ni;iy  niiikc  //  any  minilnr  of  (lines  greater  than 
tliis  valiii'  ol'  j: 

Tlun  we  may  make  ./•  any  niimhcr  of  limes  grcakr  (lian 
tills  valnu  of//. 

And  we  can  IIhh  rotiUmio,  making  oaoli  variable  oufstrip 
the  other  to  any  extent    in  a.  race  toward  intinily,  williont 

either  ever  reaeiiing  the  goal. 

• 

TiiDoitKM  III.  If  h  he  any  fi.vrd  qnmttitii,  Jioirrirr 
great,  (I nd  tc  a,  (/aantUi/  a'hir/t  ire  maij  make  as  small 
as  we  jilcase,  we  may  make  the  fJivduct  kii  less  than  ana 
ass'njnahle  qu a ntltij. 

J'roof.  If  there  is  any  smallest  value  of  /•«,  let  it  be  s. 
Because  we  may  make  «  as  small  as  we  please,  let  us  put 

Multiplying  by  k,  we  find 

k(c  <  .«?. 

So  that  k(c  may  be  made  less  than  s,  and  .v  cannot  be  the 
smallest  value. 

Def.  Tho  Limit  of  a  varialdo  quantity  is  a  valuo 
wliicdi  it  can  novor  rcacli,  but  to  which  it  may  approach 
so  jicarly  that  tho  diil'ercnco  shall  bo  loss  thau  any 
assignablo  quantity\ 

1\i:m.  In  order  that  a  variable  .\'  may  have  a  limit,  it  must 
be  a  funetion  of  some  other  variable,  and  there  must  be  certain 
values  of  this  other  variable  for  which  the  value  of  X  cannot 
be  directly  computed. 

EXAMPLES. 

I.  Tho  value  of  the  expression 

x^  —  a^ 


X  = 


X  —  a 


can  be  computed  directly  for  any  pair  of  numerical  values  of  x 
ami  a,  exce})t  those  values  Avhlch  are  equal.  If  we  supiioso 
a*  =:  a,  the  expression  becomes 


;ili'r  th.m 

iiUr  timii 

'  outstrip 
,  willioiit 

• 
hoirrrrr 
(IS  sintill 
Uidii  (lint 

et  it  be  s. 
put 


LIMITS. 


:wi 


not  be  the 


S    Jl  V.'llllO 

iH)])i()a('h 
mil  Miiy 


lit.  it  must 
li(j  certiiin 
X  cannot 


viihics  of  X 
\c  .suppose 


««-«3 


0 

0' 


a  —  a 

wliii'h,  considtTi'd  by  itsi'lf,  liiis  no  meaning. 

2.  The  t^um  of  any  tiiiite  nnmlxr  of  terms  of  a  pfoomotrloal 
progression  njay  l)i'  compnted  by  adding  tlicni.  lint  if  tiiu 
niimlier  of  terms  is  inliniti',  an  intinile  time  wonid  l)e  nipiircd 
for  the  ibrect  ealeulation,  which  id  tiu-rcfore  iinpt)ssible. 

3.  The  prca  of  a  polygon  of  any  number  of  sidc^i,  and  hav- 
ing a  given  apoincgm.  may  he  eomjtuted.  But  if  the  nnmhcr 
of  sides  hccomt's  infinite,  and  the  polygon  is  thus  changed  into 
a  circle,  the  direct  computation  is  not  practicable. 

EXERCISE. 

If  we  have  the  fraction,  A"  =  — ;  >  show  that  we  may 

fix  —  L 

make  x  so  great  that  X  shall  differ  from  .^  by  less  than  y     , 
^'''  ^^'''''  Toblouo '  '^'^^'-^  ^^'''''  i;uuJ,UU( ) '  '"^^  '^  °"  indclinitely. 

Notation  of  tlio  3Iotlio(l  of  Limits. 

304:.  Put  X,  the  quantity  of  which  the  value  is  to  he 

fouml  ; 

ar,  the  independent  variable  on  which  X  de- 
pends, so  that  X  is  a  function  of  .r; 

flf,  the  particular  value  of  x  for  which  we  can- 
not comi>ute  X\ 

Z,  the  limit  of  X,  or  the  value  to  which  it 
ai)proaches  as  x  ai)i)roaches  to  a. 

Then  the  limit  L  must  be  a  quantity  ful filling  these  two 
conditions : 

1st.  Supposing  .r  to  a])proa('h  as  near  as  we  ])lcase  to  a,  wo 
must  always  be  ahle  to  find  a  value  of  x  so  near  to  a  that  the 
difference  L  —  X  shall  become  less  than  any  assignable  quan- 
tity. 

2d.  X  must  not  become  absolutely  equal  to  L,  however 
near  x  nuiy  be  to  a. 


V  V^ 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


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2.2 


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Photographic 

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23  WEST  MAIN  STREET 

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'^^\.   ^^.\s  WnS 


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302 


LIMITS. 


'Rksi.     Tlic  quantity  a,  toward  wliioli  x  approaches,  jaiay  be 
cither  zero,  infinity,  or  some  finite  quanHty. 

Example  i.     Suppose 

„       01?  —  a^ 
2l  =  • 

X  —  a 

By  §  93,  this*  expression  is  equal  to 

7?  +  ax  +  c?,  («) 

except  when  x  =  a.     But  suppose  (J  to  be  the  difference  be- 


tween X  and  a,  so  that 


X  ■=  a  ^  cJ. 


Substituting  this  vahie  in  tlic  expression  {ci),  the  equation 
becomes 

'V3    fl'i 

3a2  +  3«(5  +  ^52. 


X  —  a 


Now  we  may  suppose  6  so  small  tliat  3^(5  +  S^  shall  be  less 
than  any  quantity  we  choose  to  assign.     Hence  Ave  may  choose 


a  value  of  x  so  near  to  a  that  the  value  of 


x-^  —  Cl"^ 
X  —  a 


sliall  differ 


from  Sa''^  by  less  than  any  assignable  quantity.     Hence,  if 


x-^ 


a^ 


then 


X  = 

X  —  a 

L  —  M\ 


'.■S 


or  3a^  is  the  limit  of  the  expression  - 


a'' 


X  —  a 


as  X  approaches  a. 


X 


Ex.  2.  The  limit  of  — ^ — -,  wiien  x  becomes  incl  'finitely 
great,  is  unity. 

For,  subtracting  ilns  expression  from  unity,  we  find  tlie 
difference  to  be 


'P 


iC  +  1 

By  taking  x  sufficiently  great,  we  may  make  this  expression 
less  than  any  assignable  quantity.     (§  303,  Th.  I.)     Therefore, 

X 


X  -\-l 
limit. 


approaches  to  unity  as  x  increases,  whence  unity  is  its 


m 


LIMITS. 


3G3 


I,  may  be 


{a) 

ircnce  bc- 

3  equation 


lull  be  less 
nay  choose 

;hull  differ 

ICC,  if 


iroacbes  a. 
nc!  'finitely 
'c  find  tlic 


expression 
Therefore, 

unity  is  its 


Notation.    The  statement  that  L  is  the  limit  of  X  as  x 
approacbes  a  is  expressed  in  the  form 

Lim.  X(x=a)  =  L. 

Tlie   conclusions  of   the  last   two   examples  may  be  ex- 
pressed tlius : 


Lim. 


x^  —  (fi 


(x=a)  =  otfi. 


X 


Lim.  — -—-  ix^r.)  =  1. 

X  -{-  1 


X  —  a 

IiEM.  This  form  of  notation  is  often  used  for  the  follow- 
ing purpose.  Having  a  function  of  x  Avhich  we  may  call  X^ 
the  form  X^^^a)  means,  "  the  value  of  X  when  x  ■=  a." 

EXAMPLES, 

{X^  +  (iyx=a)  =  (i^  +  a.  {x^  —  a^)(x^a)  =  0. 

{u^  +  2ub)(u=^-d)  =  —  t)\ 

If  we  require  the  limit  of  a  fraction  when  both  terms  be- 
come zero  or  infinite,  divide  hoth  terms  hy  some  coinmoiv 
factor  wTiicli  hccomes  zero  or  injlnitij. 

Kem.     If  the  boginnor  has  any  diifioulty  in  understanding  the  pre- 
ceding exposition,  it  will  bo  sufRoiont  for  hiin  to  think  of  the  limit  as 
simply  the  value  of  the  expression  when  the  quaiuity  oi?  which  it  t 
pends  ])ecomes  zero  or  iutinity. 

X 


For  instance, 


Lim. :r    (j  =  CO ), 

a'  +  1 


the  value  of  which  v>c  have  found  to  be  unity,  may  be  regarded  as  simply 
the  value  of  the  expression,  oo 

CO  +  1* 

Althongh  this  way  of  thinking  is  convenient,  and  gonorally  loads  to 
correct  results,  it  is  not  mathematically  rigorous,  because  neither  zero 
nor  infinity  are,  properly  speaking,  mathematical  quantities,  and  people 
are  often  led  into  paradoxes  by  treating  them  as  such. 


Find  tbc  hmit  of 


EXERCISES. 


I. 


2. 


X  ■—  CI 

— -—  when  X  approaches  infinity. 

Divide  both  terms  by  x. 
ax  4-  b 
bx  +  a 


mx" 


px^ 


ax 


when  X  approaches  infinity. 
-  when  X  approaches  infinity. 


W 


3G4 


LIMITS. 


'» 


\  —  x 
1  —  ax 
x^  —  r/2 


when  X  approaclies  infinity. 


:>' 

X  —  a 

6. 

a  -{-  X 
a  —  x 

-  when  x  approaches  a. 
when  x  approaches  infinity. 

Properties  of  Limits. 

305.  Theorem  I.  If  two  functions  are  ecfLcal,  they 
must  Jiave  the  same  limit. 

Proof.  If  possible,  let  L  and  L  be  two  different  limits  for 
the  respective  functions.     Put 

so  that  L  and  L'  differ  by  %z. 

Because  L  is  the  limit  of  the  one  function,  the  latter  may 
approach  this  limit  so  nearly  as  to  differ  from  it  by  less  than  z. 

In  the  same  Avay,  the  other  function  may  differ  from  L' 
by  less  than  z.  Then,  because  L  and  L'  differ  by  '2z,  the  func- 
tions would  differ,  which  is  contrary  to  the  hypothesis. 

Theorem  II.  The  limit  of  the  sum  of  several  func- 
tions is  equal  to  the  sum  of  their  separate  limits. 

Proof.    Let  the  functions  be  X,  X',  X",  etc. 
Let  their  limits  be     X,   Z',  L",  etc. 

Let  their  differences  from  their  limits  be  «,  a',  a",  etc. 

Then  X  =  L  —  «, 

X'  =  L'  -  a', 

Ji.        ■=.   1j      —  «   , 

etc.  etc. 

Adding,  we  have 

X+X'  +  ..r"  +  etc.  =  Z  +  X' +  X"  +  etc.  — («  4- «'  +  «"  + etc.) 

The  theorem  asserts  that  we  may  take  the  functions  so  near 
their  limits  that  the  sums  of  the  differences  «  +  a' -|- «" -f  etc. 
shall  be  less  than  any  quantity  we  can  assign. 


Hi 


LIMITS. 


365 


ual,  thcij 
limits  for 


latter  may 
less  than  z. 
r  from  L' 
,  tlie  func- 
;is. 

ral  func- 


',  etc. 


€<'-fetc.) 

ns  so  near 
h«"  +  etc. 


Let    k  be  this  quantity,  wliicli  may  be  ever  so  small ; 
n,  the  number  of  the  (|uuntities  «,  «',  a",  etc. ; 
ti,  the  largest  of  them.  • 

Because  we  can  bring  the  functions  as  near  their  limits  as 
we  please,  we  may  bring  them  so  near  as  to  make 


or        na  <  h 


Then    «-f-a'  +  «"-f-etc.  <  nu  (be<^ause  «  is  the  largest); 

whence,  «  +  «'  +  «"  + etc.  <  k. 

Therefore  the  sum  X+X'  +  X"  +  etc.  will  approach  to 
the  sum  L  ■{  L'  -^  L"  -j-  etc.,  so  as  to  differ  from  it  by  less 
than  k.  Because  this  quantity  k  may  be  as  small  as  we  please, 
Z  +  Z'  +  Z"  +  etc.  is  the  limit  of  X+  X'  +  X"  +  etc. 

Theorem  III.  TJie  limit  of  the  product  of  two  func- 
tions is  equal  to  the  product  of  their  limits. 

Proof.    Adopting  the  same  notation  as  in  Th.  II,  we  shall 

have 

XX'  =  LL'  -iiV  -  a'L  +  ««'. 

Because  L  and  L'  are  finite  quantities,  we  may  take  a  and 
«' so  small  that  aL'-\-a'L — ««'  shall  be  less  than  any  quan- 
tity we  can  assign.  Hence  XX'  may  approach  as  near  as  we 
please  to  LL' ,  whence  th    '  "tter  is  its  limit. 

Cor.  1.  Tlie  limit  of  the  product  of  any  ninnher  of 
functions  is  cqical  to  the  product  of  their  limits. 

Cor.  2.  Tlie  Umit  of  any  power  of  a  function  is  equal 
to  the  power  of  its  limit. 

Theorem  IV.  Tlie  limit  of  the  quotient  of  two  func- 
tions is  equal  to  the  quotient  of  their  limits. 

Proof.  Using  the  same  notation  as  before,  we  have  for  the 
quotient  of  the  functions, 

X' L'  —  « 

X'  -  ~L^^' 
jj 
while  the  quotient  of  their  limits  is  -j- 


366 


LIMITS. 


M 


»^t 


\0 


The  diilcrence  between  tlie  two  quotients  is 

L        U  —  a         La  —  La 
L        L  —  «        Li^L  —  «) 

IfL  is  different  from  zero,  we  may  make  the  quantities  « 

and  «'  so  snuill  that  this  expression  shall  be  less  than  any 

J  J 
quantity  we  choose  to  assign.     Therefore,  y    is  the  limit  of 

L-a'     ,,    ^.       „X'  ^ 

—f ,  that  IS,  01  -;^- 

L  ~  a  X 

30G.  Problem.  To  fuul  the  limit  of  '^^.—  as  x 
approach  "s  a. 

Case  I.     When  n  is  a  positive  ivlwlc  numler. 
We  iiave  from  §  93,  when  x  is  different  from  a, 

Q-n qu 

=  x"^-^  +  o'jif^-^  +  a^y^-^  + +  «»-!. 

X  —  (t 

Now  suppose  X  to  approach  the  limit  a.  Then  a;'*"^  will 
approach  the  limit  a^'^,  x"'-^  the  limit  a^-%  etc.  j\lultiplying 
by  a,  (fi,  etc.,  we  see  that  each  term  of  the  second  member 
ai)proaches  tlie  limit  a^-^.  Because  there  are  n  such  terms, 
we  have 

^W  __  qTI 

Lim.  — ^ {x=a)  =  7ia^~\ 

X  —  a 

Case  II.     Wlien  n  is  a  ijositive  fraction. 

7) 

Suppose  n  =^-,  p  and  q  being  wdiole  numbers.    Then 

xn  —  flfW         xQ  —  a^ 


then 
and 


X 

—  a 

X  - 

-  a 

for 

convo 

nience 

in  "wr 

iting. 

^4  = 

■y.    , 

1 
a'' 

• 

=  h', 

X  = 

■■y% 

a 

=  b'; 

xf^- 

-aP' 

v"^  - 

h^ 

y"- 
y  ~ 

b 

X  - 

-  a    ~ 

~  f- 

h"~ 

f- 

z>« 

y- 

b 

J.   y 


LIMITS. 


367 


ig 


As  .r  approaches  indefinitely  near  to  n,  and  consequently  y 
to  b,  the  numerator  of  this  ihiction  (Case  I)  ai)pn»ac'lu'S  to 
php~^  as  its  limit  and  the  denominator  to  qbi~'^.  lleiice,  the 
traction  itself  approaches  to 


Substituting  for  h  its  value  a'^,  we  have 


re"  —  «" 


p-q 


Lim.  ^^1-—^^  (j=a)  =  "^hP-^  —^a'l 


'  111 


X  —  a 


•—  7ia^~K 


Hence  the  same  formulEe  holds  when  n  is  a  positive  fraction. 
Case  III.     When  n  is  negative. 

Suppose  n  =z  — j),  j)  itself  (without  the  minus  sign)  being 
supposed  positive.     Then 

xn  _  f,n       x-P  —  a-P         _     r.(n^  —  x^'\ 

= =  x-P  a-P  I 1 

X  —  a  X  —  a  \  X  —  a  I 

x^  —  a" 


=  —  x-P  a~P 


X  —  a 


"When  X  approaches  a,  then    x-P    approaches    arP,    and 

x^  —  a^ 

approaches /)rti^~^     Substituting  these  limiting  values, 


X  —  a 
we  have 


/J.JI  ^_  /7J1 
Lim. ix=n)  =  —a-^PpaP-^  =  —jM-p-K 

Substituting  for  —p  its  value  71,  we  have 


Lim. 


xn  —  a^ 
X  —  a 


(x=a)  =  naP'-K 


Hence, 

Theorem.     Tlie  fomnulm 


Lim. 


a:^ 


a' 


ix=a) 


ncS^-'^ 


X  —  i 

is  true  for  all  values  of  n,  whether  entire  or  fractional, 
positive  or  negative. 


I 
iiiiii 

i 


308 


BINOMIAL    THEOREM 


"I 


M 


CHAPTER     V. 

THE    BINOMIAL    AND    EXPONENTIAL    THEOREMS. 


The  Binomial  Thooreni  for  all  Values  of  the 

Exponent, 

307.  We  have  shown  in  §§  171,  2G4,  liow  to  develop 
(l+a)"'  when  n  is  a  positive  whole  number.  We  have  now  to 
find  the  development  when  n  is  negative  or  fractional.    Assume 

(1  +  xy^  =  i?o  +  ^1^  +  J^i^^  +  ^^'a-'-"^  +  etc.,         {a) 

B^,  i?j,  etc.,  being  indeterminate  coefficients.  Because  this 
Cf{uation  is  by  hypothesis  true  for  all  values  of  x,  it  will  remain 
true  when  we  put  another  quantity  a  in  jilace  of  x.    Hence, 

(1  +  a)n  —  B^-\-  B^a-\-  B^a^  +  B^a^  +  etc.  {b) 

Subtracting  {h)  from  {a),  and  jiutting  for  convenience 

X  =1  +  x,        A  =  \  +  a, 

the  difference  of  the  two  equations  {a)  and  {h)  will  be 

The  values  we  have  assumed  for  X  and  A  give 


X-A  =x 


a. 


Dividing  the  left-hand  member  by  X~  A,  and  the  right- 
hand  member  by  the  equal  quantity  x  —  a,  we  have 


X^'-A'' 


x^ 


a 


~+B, 


a^ 


^  ^  x  —  a  ^  X  —  a 


+  etc. 


X-A 

Now  suppose  X  to  approach  a.  The  limit  of  the  left-hand 
member  Avill  be  7iA"'~^,  Taking  the  sum  of  the  corresponding 
limits  of  the  right-hand  member,  Ave  shall  have 

nAy-^  =  By  H-  ^B^a  +  ^B^a^  f  ^B^a^  +  etc. 

Replace  A.  by  its  value,  1  +  a,  and.  multiply  by  1  -f-  a. 
We  then  have 


, 


s 


'm 


niyOMFAL    THEOREM. 


369 


1  +  a. 


n  (1  +  (lY  =  /?!  (I  4-  ^0  +  2i?o«  (1  +  ^0  +  "5  ^'3^'^  (1  +  «) 

+  ^B^ie  (1  -f  rt)  4-  etc. 

Multiplying  tLc  cquatijii  (/v)  by  n,  wc  have 

w  (1  4-  a)"  =  wi?o  +  ^*-^>*i^'  +  «/>2^''^  +  nli^n^. 

Ef|uatinf]j  tiie  coefficients  of  the  liiic  powers  oi'  a  in  these 
Cfjuationti  (§  ;i81),  wc  have,  first, 

By  putting  a  =  0  in  equation  (//),  wc  find  /?„  =  1,  whence 

Then  we  find  successively, 
2B,  =  {n-l)  B,,  whence  B^  =  --^  B,  =  ^^-^^^• 

6b^-={n—Z)  h„y  ^3  —  — 3~^s— 17^3 

Subst"*"n.ting  these  values  of  It^,  B^,  B„,  etc.,  in  the  equa- 
tion («)  und  using  the  abbreviated  notation,  we  obtain  the 
e(iuation 

(1  +  xY  =  l  +  vx-\-  ('^)  .^2  +  {^^  y^z  ^  etc.,         {c) 
which  equation  is  true  for  ail  values  of  n. 

308.  There  is  an  important  relation  between  the  form  of 
this  development  when  n  is  a  positive  integer,  as  in  ^§  171  and 
2G-*  and  when  it  is  negative  or  fractional.  In  the  former 
case,  when  we  form  the  successive  factors  n  —  1,  n  —  'Z, 
n  —  3,  etc.,  the  n^'^  factor  will  vanish,  and  therefore  all  the 
coefficients  after  that  of  x^  Avill  vanish. 

But  if  n  is  negative  or  fractional,  none  of  the  factors 
n  —  1,  n  —  2,  etc.,  can  become  zero,  and,  in  consequence,  the 
series  will  go  on  to  infinitv.  It  tlierefore  becomes  necessarv, 
in  this  case,  to  investigate  the  convergence  of  the  development. 

If  X  >  1,  the  successive  powers  of  x  will  go  on  increasing 

indefinitely,  while  the  coefiicients  (.),  (A,  etc.,  will  not  go 
24 


370 


BINOMIAL    TIIEOIUCM. 


on  (Uminisliiiif?  iiuk'finid'ly  in  tlic  same  ralio.  For,  let  u.s 
conjsick'i-  two  succos.sivu  tcnud  of  the  duveloi)mc'ut,  the  ith  uud 
the  (/  +  1)-',  naniL'!}', 

The  quotient  of  the  .second  by  the  liivst  is 

As  i  increases  indefinitely,  this  coeflieient  of  .r  will  ajiproach 
the  limit  —1  (§  ;i04),  while  ;c  is  by  hypothesis  as  «,n-eat  as  1. 
Therefore,  by  coutinnin.ir  tlic  series,  jx  point  Avill  be  reached 
from  wliich  the  terms  will  no  lon<,an-  diminisii.     Therefore, 

Tlia  anrlopmcnt  of  (1  +  .r)«  in  powers  of  x  is  not  con- 
vergent unless  X  -C  1. 

In  consequence,  if  wo  develop  {a  +  h)n  when  7i  is  negative 
or  fractional,  Ave  must  do  so  ii)  ascending  i)owers  of  the  lesser 
of  the  two  quantities,  a  or  b. 

EXAMPLES. 

I.  Develop  (1  +  .r)^,  or  the  square  root  of  1  +  a;. 
Putting  u  =1  -,  we  have 


(I)  =  I' 


0 


;a-) 


1-2 


1-1 

2-4' 


w                  1.2.3  ~  2.4.6* 

i-3 

/'A  _  2 //A  _  1.1.3-5 

\J           4     I3/  ~  2. 4. 0-8' 


etc. 


etc. 


etc. 


i(fi  uiid 


BINOMIAL    TllKOllEM.  371 

Wlieuce, 

^^■^•"^  -^  +  ie-""^r4^  +  a^T:o^--^:j:^y^  +  etc. 

If  a;  is  a  small  fraction,  tlio  terms  in  x^,  a^»,  etc.,  will  bo 

much  smaller  than  ^.r  itself,  and  the  first  two  terms  of  the 

series  will  give  a  result  very  near  the  truth.     We  therefore  ' 
cunclude: 

Tlic  square  mot  nf  1  phis  a  snifdl  fraction  is  approA'i- 
■niatcly  equal  to  1  plus  half  that  fractluii. 

2.  To  develop  v'lO. 

We  see  at  once  that  VlO  is  between  3  and  4.    We  put  10 
in  the  form 

32  +  1  =  32(1  + -J), 

when  VlO  =  3(1  +  ]\' - 

Then,  by  the  development  just  performed, 


V     "'"9/  "^2.9        8.92"^  lfi.93 


2-9       8-92       10.9=^ 


_5 
128.94 


T-;r,  4-  etc. 


We  now  sum  the  terms  : 
1st  term,    .... 


2d  " 

3d  " 

4th  " 

5th  " 

6th  « 


Whence, 


rtn 


=   1st   4-  18, 

=  2d   -^ 

=  3d    -i 18,  .    . 

=  4th  X  —5-^72, 

=  5th  X  —  7-^90, 


1.0000000 
+   .055555*3 

—  .0015432 

4-   .0000857 

—  .0000000 
+   .0000005 


A. 


Sum  =  (1  +  ^)    =  1.0540926 
VlO  =  3  X  sum  =  3.1622778 


which  may  be  in  error  by  a  few  units  in  the  last  place,  owing 
to  the  omission  of  the  decimals  past  the  seventh. 


PI 


H|, 


872 


ICXPONENTI.  I L    TlIEOliEM. 


'I 


3.  To  (U-vclop  Vs. 

Wc  aue  that  3  is  the  nearest  whole  number  of  (lie  root.    So 
Ave  put  

V8  =  V(3'-i)  =  'y/.T-'(i-J)  =  .'i(i-,y, 

from  which  the  development  may  hu  cllected  as  before. 

EXERCISES. 

1.  Compute  the  scjuare  root  of  8  to  G  decimals,  and  from  it 
llnd  Die  stjuare  root  of  'Z  by  §  183. 

2.  Develop  (I  —  u)K 

3.  Develop  (1  —  xY^  and  exi)rcs,s  the  term  in  xK 


Torm  in  x^  =  — 


1-3.5 2t  — 1 


2-1-G  .  .  .  .'Zi 


xK 


4.  Develop 


and  express  the  general  term. 


(1  +  ^y 

/        1\"* 

5.  Develop  (1  +     I   and  express  the  general  term. 

6.  Develop  (1  —  xy'\  and  express  the  general  term. 

7.  Develop  the  m^^  root  of  1  +  wj. 

8.  Develop  {a  —  b)~%  when  a  <.h, 

9.  Develop  (1  —  a:)""*,  when  ic  >  1. 

Because  the  development  will  not  be  convergent  in  ascend- 
ing powers  of  x  when  a:  >  1,  we  transform  thus : 


and  so  put 


/  1\~^ 


10.  Develop  the  m^  power  of  1  H 


m. 


II.  Compute  the  cube  root  of  1610  to  six  decimals. 


t 


EXPONKNTIA  L    TJIh'OIih'M. 


373 


Dot.       So 


)' 


12.  Develop  (\/«  +  V/'')". 

i.^  Usiiij,'  the  functional  notation, 

^(„,)  =  i+(^;')^  +  (':>^+(';;).^'+otc., 

miiltiply  till'  two  scm-Iom,  <I>  (ni)  and  <l>(n),  and  siiow  by  t lie  for- 
niuliu  of  §  "iOl  that  tiiu  product  is  tMpiul  to  (p{in  -f  ii). 


from  it 


-f  etc. 
—  1 


xK 


ascend- 


Tlio  Exponential  Thooivin. 

JiOO.    Let  it  be  retjuircd,   if  ])ossible,  to  develop  a^  in 
powers  oH  X,  a  being  any  (puuility  whatever.     Assume 

aa'  =  Co  +  Cjo;  +  C^x*  +  C^aP  +  etc.  (1) 

to  be  true  for  all  values  of  .r.     Putting  any  other  quantity  y  in 
place  of  X,  we  shall  have 

«2/  =  (7o  +  C,ij  +  C,if  +  C,f  +  etc.  (3) 

By  the  law  of  exponents  wc  must  always  have 

Now  the  value  of  «*+y  is  found  by  writing  x  -{-  y  for  a;  in 
(1),  which  gives 

a^^y=  C^  +  C,{x-^y)-{.C^{x+ijy-\-C,  {x  +  yY  +  ctc.    (3) 

On  the  other  hand,  by  multiplying  equations  (1)  and  (2), 
we  find 


(^) 


a-ay  =  C,^  +  C,C,y  +  C,C„j/  +  C^Cgf/a    +  etc. 

+  CqC\x  +     CVa^y  +  C\  C^xf  +  etc. 

+  C'oCg:c2  -f  Ci  Cg^ca^  +  etc. 

+  C'oC'ga:^    +  etc. 

By  §  285,  the  cocflRcients  of  ail  the  products  of  like  powers 
of  ^' and  y  must  be  equal.  By  equating  them,  we  shall  have 
more  e([uations  tlian  there  are  ([uantities  to  bo  determined, 
and,  unless  these  equations  arc  all  consistent,  the  development 
is  impossible.  To  facilitate  the  process  of  comparison,  we 
have  in  equation  (4)  arranged  all  terras  which  are  homogeneous 
in  x  and  y  under  each  other. 


374 


BINOMIAL    THEOREM. 


!  ' 


B}'  putting  a-  =  0  in  (1),  wc  find 

a^  =  Co,        whence        C^  =  1.     (§  103.) 

Comparing  the  terms  of  the  first  degree  in  x  and  ij  in  (3) 
and  (-4),  we  find 

Coefficient  of  x,  C^  =  C\C^  ; 

These  two  equations  are  the  same,  and  agree  witli  C^  —  Y-, 
but  neither  of  them  gives  a  value  for  C'^,  which  must  tJierelbre 
remain  undetermined. 

Comparing  the  terms  of  the  second  degree,  we  find,  by  de- 
veloping {x  +  yf, 

C,  {x^  +  2X2/  +  y')  =  C,x'^  +  C^xy  +  C,y% 

which  gives  20 ^  —  C\% 


whence 


^''  -  1.-2  ^i'* 


Comparing  the  terms  of  the  third  order  in  the  same  way, 
we  liave 

C,{x^+3:if^y-\-dxy^^y^  =  C,a^+C,C^x^y+C^C^x7f-{-C,y% 


which  gives 
whence 


3C,  =  C,C,  =  '^C,^; 
^3  -1.2.3^^- 


If  the  successive  values  of  C  follow  the  same  law,  we  shall 
have 

ff    —  Jl.  n  4. 


and  in  general. 


C«  --  —  C  n 

'-•71    I  '-^i     • 


(5) 


Let  us  now  investigate  whether  these  values  of  C  render 
the  equations  (3)  and  (4)  identically  equal. 

Let  us  consider  the  corresponding  terms  of  the  w'^  degree, 
n  being  any  positive  integer.     In  (3)  this  term  will  be 

Cn{x-\-y)''. 


f 


I 


EXPONENTIAL    TUEOREM.      '  375 

Expanding,  it  will  be 

Cn    x^  +  nx^'-^ij  +  ('I)  x^-^y^  +  i^^  .r^-^tf  +  etc.        (C) 
In  (4)  the  sum  of  the  corresponding-  terms  Avill  bo,  putting 

CnX'^^G^  Cn-1  x^-'y-\-  C.  Cn-%x^-^yiJr  C'a 0^-3 a;"-3/  + etc    (T) 

The  first  terms  in  tlie  two  expressions  are  identical. 
Tlic  comparison  of  the  second  terms  gives 

C 
nCn  =  C\Cn-h        whence        d  —  -^  Cu-\. 


n 


This  corresponds  Avith  (5),  because  (5)  gives 


Ct-i  = 


1  ^iTl-l 


fill- 1 


and  if  we  substitute  this  value  of  Cn-i  in  the  preceding  ex- 
pression for  Cn,  it  will  become 


C-ji  — 


6f  rf 


n  (u  —  1) !        n\ 


which  ngrees  with  (5). 

The  third  terms  of  (G)  and  (7)  being  equated  give 


y .)  j  Cn  —   u  2  C,j-2- 


Substituting  the  values  of  Cn,  C„,  and  Cn-2  assumed  in  the 
general  form  (5),  we  have 

I")  JL  r''  —  1  L__  r'« 

and  we  wish  to  know  if  this  equation  is  (rue. 

Multiplying  both  sides  by  u\  and  dropping  the  common 

factor  6i",  it  becomes 

/n\  _  nl  I 

\Z/  ~  2!  (w-2)l' 
which  is  an  identical  equation. 

In  the  same  way,  the  comparison  of  the  following  terms  in 
(G)  and  (7)  give 

/«\  _  w! hi\  _         nl 


376 


EXPONENTIAL    THEOREM. 


i      '^ 


\\\   > 


all  of  which  are  identical  eqniitions.  Hence  the  conditions  of 
the  development,  namely,  that  (G)  and  (7),  and  therefore  (3) 
and  (4),  shall  be  identically  equal,  arc  all  satisfied  l^y  the  valued 
of  the  coefficients  C  in  (5).  Substituting  those  values  in  (1), 
the  development  becomes 

«^  =  1  +  C,x  +  J-  C,^x^  +  ~]~  C,h^  +  etc.        (8) 

This  development  is  called  the  Exponential  Theorem, 
as  the  development  of  {a  +  />)"  is  called  the  binomial  theorem. 

310.  The  value  of  C^   is  still  to  be  determined.    To  do 

this,  assign  to  x  the  particular  value  ^r-  Then  the  equation 
(8)  becomes  * 

'-  111 

a^.  =  1  +  1  +  --  +  -—  +  j-^-^-  +  etc.,  ad  inf.     (9) 

The  second  memlier  of  this  equation  is  a  pure  numl  r, 
■without  any  algebraic  symbol.  We  can  readily  compute  its 
approximate  value,  since  dividing  the  third  term  by  3  gives 
the  fourth  term,  dividing  this  by  4  gives  the  fifth,  etc.     Then 

1  +  1  =  2.000000 

1  -f-  1-3  =  .500000 

1  4-  1-2-3  =  .1GCGG7 

1  -r-  1-3-3-4  =  .0416G7 

1  -~  1.2-3.4.5  =  .008333 

1  -^  1.2.3.4.5.6  =  .001389 

1  -^  1.2.3.4.5.6.",  =  .000198 

1  -^  1.2.3.4.5.6.7.8  =  .000025 

1-^1.2.3.4.5.6.7.8.9=:  _^000003 

Sum  of  the  series  to  6  decimal,    2.718282 

This  constant  number  is  extensively  used  in  the  higher 
mathematics  and  is  called  the  Napenan  base.*  It  is  re})re- 
sented  for  shortness  by  the  sym])ol  e,  so  that  e  =  2.718282.... 

The  last  equation  is  'herefore  written  in  the  form 

a^«  =  e. 
*  After  Baron  Napier,  the  inventor  of  logarithms. 


i 

i 


iitlons  of 
L-eibro  (o) 
:he  valued 
es  in  (1), 


;c. 


(8) 


heorem, 

tlieoreni. 

1.  To  do 
I  equation 

inf.     (9) 

B  number, 
mpute  its 
ly  3  gives 
h.     Tiien 


higher 
IS  rei)ru- 


\ 


\ 


i 

\ 


EXPONENTIAL    THEOREM. 


377 


Raising  to  the  C\^^  power,  we  have  a  =  e^K    Hence  : 

Tlie  quantity  C^  is  the  exponent  of  the  power  to  which 
we  must  raise  the  constant  e  to  produce  the  number  a. 

We  may  assign  one  vahie  to  a,  namely,  e  itself,  which  will 
lead  to  an  interesting  result.  Putting  a  =  c,  we  have  C\  =1, 
and  the  exponential  series  gives 


e« 


X 


a? 


a.-^ 


If  we  put  x=:l,  we  have  the  series  for  e  itself,  and  if  we 
put  X  =z  —  1,  we  have 


e-i  =  -  =  1 

e 


1 1_  1 


etc. 


We  thus  have  the  curious  result  that  this  series  and  (0)  are 
the  reciprocals  of  each  other. 

EXERCISES. 

1.  Substitute  in  the  first  four  or  five  terms  of  the  expres- 
sions (0)  and  (7)  the  values  of  Cg,  C^,  Cn-2,  etc.,  given  by  (5), 
and  show  that  (6)  and  (7)  are  thus  rendered  identicjdly  equal. 

Note.  This  is  moroly  a  slight  modification  of  the  process  we  have 
actually  followed  in  comparing  the  coefficients  of  like  powers  of  ;c  and  i/ 
in  (G)  and  (7). 

2.  Compute  arithmetically  the  values  of  2.71832,  2.7183~^, 
and  2.7183~2,  and  show  that  they  are  the  same  numbers,  to 
three  places  of  decimals,  that  we  obtain  by  putting  x  =  2, 
x=  —  1,  and  X  =  —2  in  (10),  and  computing  the  first  eight 
or  ten  terms  of  the  series. 

3.  Since  c^+*  =  eef^,  the  equation  (10)  gives,  by  substituting 
the  developments  of  e^^^^  and  e*, 

(1  _|_  xY       (1  +  .r)3       (1  +  xY 


1  +  1  +  a;  + 


2! 


+ 


3! 


+ 


=  .(= 


4! 


X* 


4-  etc. 


«« 


X* 


.l  +  ^  +  2-!^-3-!+4-!  +  ^^^-;- 

It  is  required  to  prove  the  identity  of  these  developments, 
by  showing  that  the  coefl&cients  of  like  powers  of  x  are  equal. 


;  "W 


■.  i 


378 


logahitums. 


'I 


CHAPTER    VI. 

LOGARITHMS. 

311.  To  form  the  logarithm  of  a  number,  a  constant  num- 
ber is  assumed  at  i)leasurc  and  called  the  hase. 

Drf.  The  Logarithm  of  a  number  is  the  exponent 
of  the  i)ower  to  which  the  base  must  be  raised  to  \)Y0- 
duce  the  number. 

The  loirarithm  of  x  is  written  \o^  x. 


Let  us  put 


Then 


a,  the  base ; 

X,  the  number ; 

I,  the  logarithm  of  x. 

a^  =  T. 


Rem.  For  every  positive  value  we  assign  to  x  there  "will  be 
one  and  only  one  value  of  /,  so  long  as  the  base  a  remains  un- 
changed. 

Drf,  A  System  of  Logarithms  means  the  loga- 
rithms of  all  positive  numbers  to  a  given  base.  The 
base  is  then  called  the  base  of  the  system. 

Properties  of  Logiirithins. 

312.  Consider  the  equations, 
^^  =  1  ;    I 
(0-  =z  a]   \  whence  by  definition, 

«2  rr:  «2  .    ) 

Hence, 

I.   Tlie  lo^nritlnn  of  1  7,9  zero,  irhatever  he  the  base. 
II.   TJic  lo'Jaritlnih  of  the  hasc  is  1. 
III.  Tlie  logarithm;  of  any  ninnher  between  1  and  the 
base  is  a  positive  fraction. 

rV.  The  logaritlnns  of  powers  of  the  base  are  integers, 
hut  no  other  logarithms  are. 


Mogl   =0; 
1,  -j  log  «   =  1 ; 
(  log  «2  =  2. 


tant  num- 

'xponent 
I  to  j)ro- 


re  will  be 
lains  un- 

le  loga- 
e.    The 


0; 
1; 

3  base, 
'lid  the 
itegers, 


\ 


LOGAItlTlIMS. 


379 


Again  we  have 


a-i  = 


2    —   ± 


a-'  = 


^^S-    =  -1; 


a 


:i> 


«-"  = 


a 


■It, 


IIciicc, 


a 
whence  by  definition,  l  Io<?  ^    =  ~  2  • 

log  —  =   —  ;i. 


V.  r^c  logarithm  of  a  munher  hetivecn  0  azi^Z  1  is 
negative. 

Again,  as  we  increase  n,  the  value  of  a^  increases  without 
limit,  and  that  of  -  approaches  zero  as  its  limit.     Hence, 

yi.  Tlie  logarithm  of  0  is  negative  ijifijUfj/. 

VIL  TnEORE^r.     Tlte  logarithm  of  a  product  is  equal 
to  the  sum  of  the  logaritlims  of  its  factors. 

Proof,     Let  p  and  q  be  two  factors,  and  sujipose 

h  =  log  2h  h  =  log  q. 

Then  a^  =  p,  «&  =  ^. 

Multiplying,  a^a^  =  a^+f^  =  jjq. 

Whence,  by  definition, 

h-hk  =  log  (pq), 
or  log  ;?  +  log  q  z=  log  (pq). 

The  proof  may  be  extended  to  any  number  of  factors. 

VIII.  Theorem.  TJiclogarithm,  of  aquotientis  found 
hysuhtracting  the  logarithm  of  the  divisor  from  that  of 
the  dividend. 

Proof.  Dividing  instead  of  multiplying  the  equations  in 
the  last  theorem,  we  have 


^  h  v      P 


380 


LOOABITIIMS. 


I'         ' 
1     < 


ii 


mi 


or 


Hence,  by  dcfiniton,        h  —  k  =  log  - , 
log;;— log<7  =  log^. 


IX.  Theorem.  Tlic  Ingaritlun  of  any  power  of  a  jnini- 
hcT'  is  equal  to  the  logarWiDi  of  the  nuw.hcr  DiultipUcd 
hij  the  exponent  of  the  poiver. 

Proof.    Let  h  =  log  p,  and  let  7i  be  the  exponent. 

Then  ft'  =  p. 

Raising  both  sides  to  the  w/'"'  power, 

Whence  nh  —  log;;", 

or  71  log  ;;  =  log  p^. 

X.  Theorem.  Hie  logarithm  of  a  root  of  a  number 
is  equal  to  tlie  logaritliiiv  of  the  number  divided  by  the 
index  of  the  root. 

Proof.  Let  s  be  the  number,  and  let  p  be  its  n^^  root,  so 
that 

p  =z  \^s        and        s  =  ;;". 

Hence,  log  s  =  log  jj''  =  n  log  jJ-     (IX.) 


Therefore, 


or 


logs 
log  p  =  —  — , 

,  y  log  S 

log  ys  =  — - — 


_  A 


Note.    We  may  also  apply  Th.  IX,. since  p  =  sK     Con 
sidering  -  as  a  power,  the  theorem  gives 

log  p  ■=  -  log  s. 


EXERCISES. 

Express  the  following  logarithms  in  terms  of  log;?,  log  q^ 
log  X,  and  log.?/,  a  being  the  base  of  the  system : 


i 


) 


tlti  piled 


t. 


number 
1  by  tha 

^  root,  so 


A 


LOQARITUMS.  381 

1.  L<g  p%  Ans.  2  \ogp  4-  log  q. 

2.  Log  pn^. 

3-  Logyr<jr5.  ^,  Logpfj^i/^. 

5.  Log  -  =  log  ^^;-^,  and  explain  the  identity. 

6.  Log  —  =  log  xyjr^  Q~^- 

Ans.    Log  X  +  log  y  —  log ])  —  log  g*. 


7.  Log  -^'-. 


^'    Log  „«„3 


p"'f 


9.  Log  a/.^  (Xotc,  §  123).      10.  Log  Vx  y/y. 


II.  Los: 


VI- 


13.  Log  ax. 


12.  Log  Vfl!. 

X 

14.  Log 


« 


a: 


15.  Log  — 


a" 


16.  Log  -^ 


(,npm 


«    :.^y3 


17.  Log  ^/~(^-  ■  x\ 

18.  Log  Vl  —  a^. 


Ans    Lo^  (^^  +  •^')  +  ^og  («  —  ^') 
19.  Log  (rt^  _  2-2). 


EX.) 


:*.     Con- 


i'j  log  (Z, 


1 

1 


Logarithmic  Series. 

313.  Rem.  The  logarithm  of  a  number  cannot  bo  devel- 
oped in  powers  of  the  number.     For,  if  possible,  suppose 

log  2;  =  (7o  +  Cja:  +  C^x^  +  etc. 
Supposing  a:  =  0,  we  have 

C\  =  log  0, 

which  we  have  found   to   be  negative   infinity   (§  312,  YI). 
Hence  the  development  is  impossible. 

But  we  can  develop  log  (1  +  y)  in  powers  of  y.  For  this 
purpose,  we  develop  (1  +  y^'^  by  both  the  binomial  and  expo- 
nential theorems,  and  compare  the  coefficients  of  the  first 
power  of  X.    First,  the  binomial  theorem  gives 

(1  +  y)-  -^l  +  xy  +  5^i^D  y^  +  :ii:!jZ.li(|ZL?)  ,^3+etc. 


i! 


382 


LOOARITUMS. 


i 


If  wo  develop  the  cocfTicients  of  if,  if,  etc.,  by  performing 
the  multiplicatioiiij,  we  have 


part  in  x  = 


X 

2 


(C         i(     y^ 


In  general,  in  the  coefficient  of  //",  or 

x{x—l){x  —  '-Z).,..  {x  —  n  +  1), 
the  term  containing  the  first  power  of  ^  will  be 


Hence, 


±1-2-3 {)i  —  1) .r  _       X 

l'"Z-'i  . . . .  n  ~       n 


(1  +  yY  =z  1  +  X  \p  —  •^-  +  '!-  —  |-  +  etc. j  +  terms  in  x\  ofi,  etc. 

On  the  othci  hand,  the  exponential  development,  §  309,  (8), 
gives,  by  putting  1  +  ?/  lor  «. 

(1  +  yY  =  1  4-  C\x  -\-  terms  in  x\  x?,  etc. 

Equating  the  coefficients  of  x  in  these  two  identical  series 
we  have 

/4 


c.=y-t^i 


r 


+  etc. 


0) 


Tlie  value  of  C^  is  given  by  the  theorem  of  §  310,  putting 
1  -\-  y  for  a;  that  is,  (7^  is  here  defined  by  the  equation 

eO.  =  I  ^  y. 

Hence,  if  we  take  the  number  e  (§  310)  as  the  base  of  a 
system  of  logarithms,  we  shall  have 

C,  =  log  (1  +  y). 

Comparing  with  (1),  we  reach  the  conclusion: 

TiiEOiiEM.    Assuming  the  Xapcrian  base  e  as  a  hase, 
ive  have 

log  (1  +  2/)  =  2/  -  I  +  I  -  |-  +  etc.,  ad  inf.       (2) 


LOGJIUTiJMS. 


rforming 


X 

2 


X 

—  • 

3 


X^,  7^,  etc. 

^  309,  (8), 

ical  series 

(1) 

),  putting 
ion 

base  of  a 


s  a  hase, 


»)o. ) 

erian 


I><f.     Lonraritlims  to  tlie  l:.as<^  e  aro  called  Nape 
Logarithms,  or  Natural  Logarithms. 

Hem.     The  series  (2)  is  not  convergent  when  v  >  l,  ;uul 
therefore  must  be  tran.sfurnied  I'ur  use. 

Putting  _  >/  for  y  in  (;>),  we  have 

Subtracting  this  from  (2;,  an]  noticing  that 
log  (1  +  y)  -  log  (1  -  2/)  =  log  Lt?/  (Th.  VIII), 


1  +  .y 


.e  have         log  ^5|  =  2^  +  f  +  ^  +  etc.  (.) 

Now  w  being  any  number  of  which  we  wish  to  investigate 
the  logarithm,  let  us  suppose  ?/  =  --i-T-    This  will  give 


1  +y  _  n  +  1 


it 


Whence     log  \±-l  =  k.  ''±1  =  kg  (;,  +  i)  _  kg  ,,. 
Substituring  these  values  in  (3),  we  have 


log  {n  +  1)  -  log  71  = 


2 


"^  'i  {•> 


2 


2)1  +  1^3  {2)1  +  1)3  +  5  {2n  +  1)' 
+  etc.  (4) 

This  series  enables  us  to  find  log  (w  +  1)  when  we  know 
log  71.  To  find  log  2,  we  put  n  =  1,  which,  because  log  1 
=  0,  gives 

log  3  =  2  (~  +  -i  -  +  J:_  +  J_  _L  ptp  \ 
°  \3  ^3-33  ^5.35^7.37  +  ^'^  7 

Summing  five  terms  of  this  series,  we  find 

log2  =  0.G93U7 


384 


:0OAIiITJLVS. 


riitiiii 


1  f      s- 

11; 


log  3  =  l.()98(Jl:e. 
log  9  =  2  \i)i 


:.197224. 


'uiliiig  71  =  2  in  (4),  wc  have 

loR  I!  =  l„g  3  +  2  (y  +  .^^  +  ji-,  +  ~  +  etc.). 

wliich  gives 

Because  9  =  3^, 

Putting  n  =  9  in  (4),  wc  liiive 

log  10  =  log  9  +  2  (j^  +  ^^  +  gJ-js  +  etc.), 

whence  log  10  =  2.'3025So. 

In  this  way  tlie  Naporian  logaritlimsof  all  numbers  maybe 
comi)utcd.  It  is  only  necessary  to  compute  the  logarithms  of 
the  ju'ime  nnmbers  from  the  series,  because  those  of  the  com- 
posite numbers  can  be  formed  by  adding  the  logarithms  of 
their  prime  factors.     (§  312,  VII.) 

314.  Definitive  Form  of  the  Exponential  Series.  "We  are 
now  prepared  to  give  the  exponential  series  (§  309,  8)  its  defi- 
nite form.     Since  the  coefficient  C\  is  defined  by  the  equation 

e^''  =  a, 
the  quantity  C  is  the  Naperian  logarithm  of  a.     Ilence,  tho 
exponential  series  is 

ax  =  1  +  6Ll2g^  +  (^|S.52  +  <PJ^^J^  +  etc., 

which  is  a  fundamental  development  in  Algebra. 

By  putting  a  =  e,  we  have  log  a  =  1,  and  the  series  be- 
comes that  for  e^  already  found. 

By  putting  x  =  l,  we  have  an  expression  for  any  number 
in  terms  of  its  natural  logarithm,  namely, 

«  -  1  +  -3--  +  — ^ r-  +  — 3j—  +  —jr-  +  etc. 


M 


Comparison  of  Two  Systems  of  Logarithms. 

315.  Put        e,  the  base  of  one  system  ; 
a,  the  base  of  another; 
w,  a  number  of  which  we  take  the  logarithm 
in  both  systems. 


LOaAllITHMS. 


385 


}tc.j, 


cj, 


Ts  may  be 

irithius  of 

the  coni- 

rithms  of 

,  "Wo  are 
8)  its  (lefi- 
3  equatiou 

Icnco,  tlio 

-  etc., 

series  be- 
y  number 

-|-  etc. 
Itlims. 


logarithm 


Pnftin;:;  1  and  /'  for  the  lo;:fiirillnns  In  llio  two  system-,  no 

liUVU 


(*  =  n, 


«* 


n. 


aud  therefore  i^  =  a^'.  (1) 

Now  put  k  for  the  logarithm  of  a  to  the  base  c.    Tlicn 


c*  =  a. 


and  raising  both  members  to  the  l'^^  power, 
Comparing  with  (1),       I  =  kl\ 


or 


I'  ^lyc 


1 


(•2) 


This  equation  is  entirely  independent  of  w,  and  is  therefore 
the  same  for  all  values  of  n.     Hence, 

Theorem.  //  wc  DiuUiphj  the  Jo^aj'it7n)i  of  any 
iiUDibrr  to  the  h'ise  a  htj  the  logdritliin  of  a  to  the  base  e, 
we  shall  have  the  lo'JariUun  of  the  mcDiher  to  the  base  e. 

31G.  Althouo;h  there  may  be  any  number  of  systems 
of  logaritlinis,  only  two  are  used  in  practice,  namely  : 

1.  The  natural  or  Naperian  system,  base  =  e  = 
2.718282  .... 

2.  The  common  system,  base  =  10. 

The  natural  system  is  used  for  purely  algebraic 
purposes. 

The  common  system  is  used  to  facilitate  numerical 
calculations. 

Assigning  these  values  to  e  and  a  in  tho  preceding  section, 
the  constant  k  is  the  natural  lognrithm  of  10,  which  we  have 
found  to  be  2.302585. 

Therefore,  by  (2),  for  any  numV)er, 

nat.  log  =:  common  log  x  2.302585, 


and 


nat.  loff. 
common  Icj  =  -^~-^  r:^ 


llencc, 


=  nat.  log  X  0.4342914.... 


25 


380 


LOGAIHTUMS. 


'I 


TilEOUMM.  Ilir  rnmwoti  logarithm  of  nvy  nunihrr 
7)}/nf  he  found  bij  niuiti pi (liiuj  ifs  nafuraf  lo'^ai'lth in  Inj 
().i;{4'^!»44  ....  oi'  by  the  rcviprocal  of  the  Aa/trrinn  In^n- 
ritlini  of  U). 

Jhf.  Tlio  ninnl)(M"(M:Mi>!)44  is  called  the  Modulus 
of  tliu  coimnou  ayistom  ot'loguritliiua. 

EXERCISES. 

1.  Show  (hilt  if  a  and  b  ho  any  two  l)asos,  the  lo^rarithm  of 
a  to  the  hjLso  b  ami  tho  io<,'arithni  of  /;  to  tlio  hasu  a  arc  tiie  re- 
ciprocals of  each  other. 

2.  What  (loos  this  tlioorom  express  in  the  case  of  the  natu- 
ral and  common  systems  of  lo<;ariLhnis  ? 

Coiiiinoii  Lo^iiriihins. 


wc  have  to  hase  10, 


loor 

100 

— 

2, 

log 

10 

=r 

1, 

log 

1 

= 

0, 

log 

1 

10 

= 

-1, 

log 

1 

nr 

(lij  ilf ' 


etc. 

The  following  conclusions  respecting  common  logarithms 
will  he  evident  from  an  inspection  of  the  ahovo  examples: 

I.  Tlie  lo^dvitlnU'  of  (tny  nuniher  betiuecii  1  (tnd  10  is 
a  fraction  between  0  and  1. 

II.  Tlie  lo'^aritlun  of  a  number  with  two  dibits  is  1 
plus  some  fruction. 

III.  In  gcjieral ,  the  logaritlim  of  a  nmnber  of  i  digits 
is  i  —  1,  plus  some  fraction. 

IV.  The  logarithm  of  a  fraction  less  than  unity  is 
negative. 

V.  77/ r  logarithms  of  two  numbers,  the  reciprocal  of 
each  other,  are  equal  and  of  opposite  signs. 


in    ? 


LOUAlilTllMS. 


387 


'  nnvthrr 
ritlun  hif 

Hill  h><J(i- 

Modulus 


p:arithm  of 
lire  the  ro- 

£  the  luiLu- 


=       2, 

=       1, 
=       0, 

=  -1, 


k'tc. 
lo^iU'itlims 

(tiul  10  is 
di'^its  is  1 
iofi  dibits 
hi  iiiiiiy  is 
fprocal  of 


VT.  //  one  innuhcr  is  10  times  another,  its  ht^m-itJini 
it'ill  In'  greater  Inj  unity. 

Vrmf.  If  lO/  =  w, 

lluM  10'*'  =  10  X  lev  =  10/<. 

llenco,  if  I  =  log  w, 

tlK'ii  ^  4-  1  =  log  lO/i. 

IU8.  To  give  mi  itloii  of  llio  pmgrossion  of  logarithms,  the 
following  ttibie  of  logurithiiLS  of  tho  lirst  II  nuiuhors  mIiomIiI  bo 
stiiiticd. 

Tliu  logarilhins  arc  not  exact,  bocjiusc  all  logarithms,  ex- 
cept those  of  powers  of  10,  are  irrational  mmilK'rs,  and  there- 
fore when  expressed  as  decimals  extend  out  indelinitely.  We 
give  only  the  lirst  two  decimals. 

log  1  =  0.00,  log  7    =  0.8r), 

log^  =:  0.30,  logs    —  0,00, 

log;3  =  0.48,  logo     =  0.05, 

log  4  =  O.GO,  lug  10  =  1.00, 

log  5  =  0.;0,  log  11  =  1.04. 
log  0  =  0.78, 

It  will  be  noticed  that  tho  difference  between  two  consecn- 
tive  logarithms  continually  diminishes  as  the  numbers  increase. 
For  instance,  the  dilFerence  between  log  20  and  log  10  must 
by  §  312,  VIII,  be  the  same  as  between  log  1  and  log  2. 

819.  Compntcdion  of  Lofjnrithms.  Since  the  logarithms 
of  all  composite  numl)ers  may  be  found  by  adding  the  loga- 
rithms of  their  factors,  it  is  only  necessary  to  show  how  the 
logarithms  of  prime  numbers  are  computed.  V»e  have  already 
shown  (§  313)  how  the  natural  logarithms  may  be  com})utetl, 
and  (§  31 G)  how  the  common  ones  maybe  derived  from  them 
by  multiplying  by  the  modulus  0.4342!)44....  It  is  not  how- 
ever necessary  to  multiply  the  whole  logarithm  by  this  fuelor, 
but  we  may  proceed  thus: 

We  have,  putting  J/ for  the  modulus, 

com.  log  n  =  M  nat.  log  n, 

com.  log  {n  -\-  \)  =  M  nat.  log  {n  +  l)-, 


V. 


388 


LOGAIUTUMS. 


M 


whence,  by  subtraction, 

com.  log  {n  +  ^ )  —  com.  log  n  =  3f  [nat.  log  {n  + 1) — nat.  log ;/] ; 

aiul,   by  substituting    for     nut.  log  {n  +1)  —  nat.  log  u     its 
value,  §  313, 

com.  log  (n  -f-  1)  =  com.  log  7i  -f  2Jf 


+ 


jZu  +  1 
1 


+ 


d{2H  +  1)3 

^  +  etc. 


5  {2ti  +  1) 

By  means  of  this  series,  the  computations  of  the  successive 
logarithms  may  be  carried  to  any  extent. 

Tables  of  the  logarithms  of  numbers  up  100,000,  to  seven  places  of 
decimals,  are  in  common  use  for  astronomical  and  mathematical  calcula- 
tions. One  table  to  ten  decimals  was  published  about  the  beginning'  of 
the  present  centur\ .  The  most  extended  tables  ever  undertaken  were 
constructed  under  the  auspices  of  the  French  government  about  1795,  and 
have  been  known  under  the  name  of  Lcs  Grandcs  Tables  du  C/uhidre. 
Many  of  the  logarithms  were  carried  to  nineteen  places  of  decimals. 
They  were  never  published,  but  are  preserved  in  manuscript. 

320.  It  may  interest  the  student  Avho  is  fond  of  computa- 
tion to  show  how  the  common  logarithms  of  small  numbers 
may  be  computed  by  a  method  based  immediately  on  first 
principles. 

P 

Let  n  be  a  number,  and  let  -  be  an  approximate  value  of 


its  logarithm. 


"We  shall  then  have. 


p 

n  =  10?, 

or,  raising  to  the  q^^  power, 

n^  =  IQP. 

Hence,  could  we  find  a  power  of  the  number  which  is  also 
a  power  of  10,  the  ratio  of  the  exponents  would  at  once  give 
the  logarithm.  This  can  never  be  exactly  done  with  whole 
numbers,  but,  if  the  condition  be  approximately  fulfilled,  we 
shall  have  an  approximate  value  of  the  logarithm. 

Let  us  seek  log  2  in  this  way.  Forming  the  successive 
powers  of  2,  we  find 

2^0  =  1024  =  103(1.024).  (1) 


11  at.  log  w]; 


log  u     its 


'd{2H  +  1)3 

+  etc. 

)  successive 


von  places  of 
tical  calcula- 
beginning  of 
ertakcn  wore 
nit  1795,  and 
du  Cddiidre. 
of  decimals. 


f  compnta- 
'1  numbers 
ly  on  first 

te  value  of 


icli  is  also 
'j  once  give 
tvith  whole 
ulfilled,  Ave 

successive 


(1) 


L0GAIUTIIM8. 


389 


Hence,  3  :  10  =  0.3  is  an  approximation  to  log  2.  To 
find  a  second  approximation,  we  form  the  powers^'of  l.OU 
until  we  reach  a  number  nearly  equal  to  2  or  10,  or  the  quo- 
tient of  any  power  of  2  by  a  power  of  10.  Suppose,  for  instance, 
that  we  find 

1.024*  =  2. 

Because  1.024  =  2^  -~  10^,  this  equation  will  give 

'  2**^\^ 


(, 


\mf  ~^'     or    2i<^  r=2.10*",    or    2io-«-i  =  103«, 
which  will  erive 


log  2  =  —^ — 
"^  10.«  — 1 


If  we  form  the  powers  of  1.024  by  the  binomial  theorem, 
or  in  any  other  way,  avc  sIuiU  find  that  x  is  between  29  and  30^ 
from  which  we  conclude  that  log  2  =:  0.301  nearly. 

To  obtain  a  yet  more  exact  value,  we  form  the  30th  power 
of  1.024  to  six  or  seven  decimals,  and  put  it  in  the- form 

1.02430  =  2  (1  +  «), 
where  «  will  be  a  small  fraction. 

Then  avc  find  what  power  of  1  +  «  will  make  3.  Let  y  be 
this  power.  Eaising  the  last  equation  to  the  yth  power,  we 
have 

1.024%  =  2y(l  4- «)y  =r  2^+1. 
Putting  for  1.024  its  value,  2io  -v- 103,  this  equation  becomes 

=  2y+»,        or        22%-i  =  lO^oy, 


10% 


whence, 


log  2  = 
"^  299^ 


2299y-l 

00;/ 


By  a  little  care,  the  value  of  y  can  be  o])tained  so  accurately 
that  the  value  of  log  2  shall  be  correct  to  8,  9,  or  10  places  of 
decimals. 

The  power  to  which  we  must  raise  1  +  «  to  produce  2  will 
be  approximately     '^^''  "'^",  when  «  is  very  small. 


390 


LOGARITHMS. 


M\ 


M  (if 


EXERCISES. 

1.  11.  the  common  system  {a  =  10)  we  have 

log  2  =  0.30103,  log  3  =  0.47712. 

ITciice  find  the  logarithms  of  4,  5,  6,  8,  9,  VZ,  12|,  15,  16, 
IGf,  18,  20,  250,  0250. 

Note  that  5  =  V,  12|  =  J-g".  16|  =  J-O",  and  apply  Tb.  VIIL 

2.  How  many  digits  are  there  in  the  hundredth  power  of  2? 

3.  Given  log  49  =  l.G9010'5 ;  find  log  7. 

4.  Given  log  1331  =  3.124178;  find  log  11. 

5.  Find  the  logarithm  of  105  and  1.05  from  the  above  data  ? 

6.  Find  the  logarithm  of  1.0510. 

7.  If  $1  is  put  out  at  5  per  cent,  per  annum  compound 
interest  for  1000  years,  how  many  digits  will  be  required  to 
exi)ross  the  amount?     (Compare  §  21G.) 

8.  Prove  the  equation 

log  a;  =  ^  log  {x  +  1)  +  ^  log  {x  —  1) 


+  i/ 


+  o 


+ 


+  etc. 


_2x2  -  1    '   3  (2a.-2  —  1)3  ^  5  {2xi  —  If 

9.  If  «/  =  log  n,  of  what  numbers  will  y  +  '!i,  y  -\-  '^,  y  —  1, 
and  y  —  2  be  the  logarithms  ? 

10.  Find  X  from  the  equation  cf^  =  li. 

Solution.     Taking  the  logarithms  of  both  members,  we  have 

a;  log  c  =  log  h\ 

log  h 
log  c 

1 


whence, 


X 


II.     cfl^  —  n. 


12.     c*a;  = 


m 


14.     b-^  =  p. 


13.     If^  =  -. 
P 
Show  that  the  answers  to  (13)  and  (14)  are  and  ought  to  be  identical. 

15.      aca;  —  ^^.  16,      bd^  —  k. 

17.     Find  X  and  y  from  the  equations 

a^hv  =  py  (Oilf^  =  q. 


m 


15,  IG, 


rerof2? 


BOOK     XII. 
IMA  G  IN  A  RY     QUANTI  TIE  S, 


ve  data  ? 

mpound 
uired  to 


+  etc. 


dentical. 


CHAPTER    I. 

OPERATIONS    WITH    THE    IMAGINARY    UNIT.* 

331.  Since  the  square  of  cither  a  negative  or  a  positive 
quantity  is  always  positive,  it  follows  that  if  we  have  to  extract 
the  sfjiiare  root  of  a  negative  quantity,  no  answer  is  possible, 
in  ordinary  positive  or  negative  numbers  (§§  ITO,  200). 

In  order  to  deal  with  such  cases,  mathematicians  have  ])een 
led  to  suppose  or  imagine  a  kind  of  numbers  of  which  the 
squares  shall  be  negative.  These  numbers  are  called  Imagi- 
nary Quantities,  and  their  units  ;M-e  called  Imaginary 
Units,  to  distinguish  them  from  the  ordinary  positive  and 
negative  quantities,  which  are  called  real. 

323.  TJie  Imaginary  Unit.  Let  us  have  to  extract  the 
square  root  of  —  9.  It  cannot  be  equal  to  +  3  nor  to  —  3, 
because  the  square  of  each  of  these  quantities  is  +  9.  We 
may  therefore  call  the  root  V —  0,  just  as  we  put  the  sign  \/ 
before  any  other  (piantity  of  which  the  root  cannot  be  extracted. 
But  the  root  may  be  transformed  in  this  way  : 

Since  —  9  =:  -|-9  x  —1, 

it  follows  from  §  183  that 

V-^  =  Vo  V~i  =  n\/~r. 

*  It  is  not  to  be  expected  that  a  bejiinncr  will  fully  understand  this 
Bubject  at  once.  But  he  should  be  drilled  in  the  mechanical  process  of 
operating  with  imagiuarics,  even  though  lie  does  not  at  first  understand 
their  significance,  until  the  subject  becomes  clear  through  familiarity. 


till 


392 


IM AG  IN  ART  QUANTITIES. 


M 


Def.    The  surd  V— 1  is  tlio  Imaginaxy  Unit.    Tlie 
imaginary  unit  is  commonly  expressed  by  the  symbol  i. 

This  symbol  is  used  because  it  is  easier  to  write  i  than 

Tlie  unit  i  is  a  supposed  quantity  such  that,  when  squared, 
the  result  is  —  1. 

That  is,  i  is  defined  by  the  equation 


fi  =  - 1. 


Theorem.  TJic  square  mot  of  any  negative  quantity 
may  he  expressed  as  a  number  of  imaginary  units. 

For  let  —  01  be  the  number  of  which  the  root  \&  required. 

Then  V—  n  =  V  +  n  V—  1  =  Vni, 

Hence, 

To  extract  the  square  root  of  a  negative  quantity, 
extract  the  root  as  if  the  quantity  were  positive,  and 
affix  the  symbol  i  to  it. 

323.  Complex  Quantities.  In  ordinary  al;.'ebra,  any  num- 
lx.'r  may  be  supposed  to  mean  so  many  units.  7  or  a,  for 
example,  is  made  up  of  7  units  or  a  units,  and  might  be  writ- 
ten 7-1  or  al. 

When  we  introduce  imaginary  quantities,  we  consider  them 
as  made  up  of  a  certain  number  of  imaginary  unite,  each  rcpro- 
sented  by  the  sign  i,  just  as  the  real  unit  is  represented  by  the 
sign  1.     A  number  b  of  imaginary  units  is  therefore  writttjn  hi. 

X  sum  of  a  real  units  and  b  imaginaiy  units  is  written 

a  4-  bi, 
and  is  called  a  complex  quantity.     Hence, 

Dp/,  a  Complex  Quantity  consists  of  the  sum  of 
a  certain  number  of  real  units  plus  a  certain  number  of 
imajjcinary  units. 

Def.  When  any  expression  containing  the  symbol 
of  the  imaginary  unit  is  reduced  to  tlie  form  of  a  com- 
plex quantity,  it  is  said  to  be  expressed  in  its  Normal 
Form. 


IMA  GINAR  Y  Q  UANTITIE8. 


393 


Fnit.    Tlie 
symbol  /. 

rite  i  than 
en  sfjuarcd, 


!  qudutity 
inits. 

\  required. 


quantify, 
itive,  and 

,  any  num- 
7  or  a,  for 
lit  be  writ- 

Kider  them 
ac'h  repro- 
ted  by  the 
written  hi, 
rittea 


e  PTllTl  of 

umber  of 

sjTTibol 
f  a  corn- 
Normal 


I 


Addition  of  Complex  Expressions. 

334.  The  algebraic  operations  of  addition  and  subtraciion 
are  performed  on  imaginary  (quantities  according  to  nearly  Ihe 
same  rules  which  govern  the  case  of  surds  (§  181),  the  surd 
being  replaced  by  i.     Thus, 

aV—  1  +  bV—  1  =  (ti  +  hi  =  {a -\- h)  i. 

Hence  the  following  rule  for  the  addition  and  subtraction 
of  imaginary  quantities : 

t'ldd  or  suhtract  all  the  real  terms,  as  in  ordinary 
al^cJjra.  Tlicii  add  the  cocfjlcicnts  of  tJie  imaginary 
unit,  and  affix  the  symbol  i  to  tlieir  sum. 

Example.  Add  a  +  bi,  G  -j-  7/,  5  —  10/,  and  subtract 
da  —  2bi  4-  z  from  the  sum. 

We  may  arrange  the  work  as  lollows: 

a  +    bi 

6  4-    71 
5  -  lOi 
—  z  —  da  4-  2bi     (sign  changed). 

Sum,      —  z  —  2a  +  11  +  (3^*  —  3)  i. 

EXERCISES. 

1.  Add  3.T  +  iyi  +  m,  2m  +  5. a,  Cnn  —  Ctiji. 

2.  Add  4ai,  17/,  da  +  Gbi,  x  ■\-  yi. 

3.  From  the  sum  a  +  bi  -\-  m  —  ui  —  p  -f  qi  subtract  the 
sum  -{-  yi  —  z  —  ui. 

Reduce  to  the  normal  form : 

4.  a  4-  bi  —  {ni  —  ni)  —  {x  4-  yi). 

5.  Ill  {a  —  bi)  —  n  {x  —  /;/). 

3Iultipliciition  of  Complex  Quantities. 

335.  Theorem.  All  the  even  powers  of  the  imagi- 
nary unit  are  real  units,  and  all  its  odd  powers  are 
imaginary  units,  positive  or  nec>ative. 


394 


IMAGINARY  QUANTITIES. 


M 


Proof.  The  imaginary  unit  is  by  delinition  such  a  symbol 
as  when  squared  will  make  —  1.     Hence, 

t"2  =  —  1. 

Now  multiply  both  sides  of  tl.is  equation  by  i  a  number  of 
times  in  succession,  and  substitute  for  each  power  of  i  its  value 
given  by  the  preci'ding  e(|uation.     We  th(!n  liiivo 

i^  z=.  —  /, 

«'•*  =—/'-=  +1  (because  i^  =  —  1), 

i^  =  —  i^  =  -f  /, 

^•0  =  —  i«  ^  4-  /2  =  _  1, 

r'  =  —  i^  =  -}-  v^  =  —i, 
etc.      etc.  etc. 

It  is  evident  that  the  successive  powers  of  i  will  always 
have  one  of  the  four  values,  i,  —  1,   —  /,  or  +  1. 


h 

i\ 

i', 

etc.. 

will  be  equal  to 

*; 

l\ 

i% 

«io. 

etc., 

((                   a 

-1; 

l\ 

i\ 

i'\ 

etc.. 

(t                   (t 

—  i; 

i\ 

i\ 

i'\ 

etc.. 

((                   ft 

+  1. 

- 


We  may  express  this  result  thus : 
//  71  is  any  integer,  then: 


To  multiply  or  divide  imaginary  quantities,  we  proceed  as 
if  they  were  real  and  substitute  for  each  power  of  i  its  value  as 
a  real  or  imaginary,  positive  or  negative  unit. 

Ex.  I.     Multiply  ai  by  xi. 

By  the  ordinary  method,  we  should  have  the  product, 
axi^.     But  i*^  =  —  1.     The  product  is  therefore  —  ax. 

That  is,  ai  x  xi  =  —  ax. 

Ex.  2.  JNIultiply  a  +  bi  by  m  +  7ii. 

ni{a  +  hi)  =  ani  —  hi  (because  nixbi  =  — hi) 

m  {a  H-  bi)  =  bmi  +  am  _ 

{in  +  Hi)  {a  +  bi)  =  am  —  bn  -\-  {an  +  bin)  i, 
which  is  the  product  required. 


a  symbol 


umber  of 
its  vtiliie 


11  always 


—  I. 

oceed  as 
alue  as 


n-oduct, 


-hn) 


^ 


IMAGINARY  QUANTITIES. 


EXERCISES. 


395 


Multiply 

I.     X  -\-  yi  uy  a  —  b.  2.     m  +  ni  by  ai, 

3.     m  —  ni  by  bi.  4.     1  -|-  ?'  by  1  —  i. 

5.  X  —  yi  by  a  +  hi.  6.     x  —  yi  by  x  -}-  yt. 

7.  rt  —  ai  —  bi  by  a  +  ai  +  bi. 
Develop 

8.  {a  +  bif.  9.     (//i  +  nif. 
10.    (1  +  0^-                       "•    (1  —  0^- 

32G.  Imaginary  Factors.  Tbe  introduction  of  imaginary 
units  enables  us  to  faetor  expressions  wbicli  are  prime  wben 
only  real  factors  are  admitted.  Tbe  following  arc  tbe  princi- 
pal forms : 

«2  +  J2  —  (^  ^  hi)  (^a  —  bi), 

oi  —  yi±  2abi  =  {a  ±  bi)\ 

Tbe  first  form   sbows  tbat  the  sum  of  two  squares  can 
always  be  expressed  as  a  product  of  two  complex  factors. 
For  example,  17  =  4^  +  l^  =  (4  +  i)  (4  —  i). 

EXERCISES. 

Factor  the  expressions : 

I.  ara  +  4.  2.     x^ -\- 2. 

3.  :^  _  o.^  +  5  =  {x  —  1)2-1-4. 

4.  x^  —  4:X  4-13  5.     «  +  5. 

6.  a-  4-  2a n  +  o>A  7.     x"^  +  2xy  +  2y^. 

327.  Fundamental  Principle.  A  complex  quantity 
A  -\-  Bi  cannot  bz  equal  to  zero  unless  we  have  both 

A  =  0        and        B  =  0. 

Proof.  If  A  and  B  were  not  zero,  the  equation  A  +  Bi  =  0 
would  give 


I  = 


A 

B' 


that  is,  the  imaginary  unit  equal  to  a  real  fraction,  which  is 
impossible. 

Cor.    If  both  members  of  an  equation  containing  imagi- 


306 


IM AG  IN  ART  QUANTITIES. 


h 


'\ 


iiiiry  units  nro  reduced  to  the  normal  form,  so  that  the  equation 

sJKiil  be  ill  the  form 

A^  ni  =  il/+iV7, 

we  must  have  tiie  two  equations, 

A  =  M,  B  =  N. 

For,  l)y  transposition,  we  obtain 

A  -M->r{B-N)i  =  0, 

whence  the  theorem  gives  A  —  M=  0,  B  —  N=  0.     Hence, 

I'A'crij  cqiuitloii  between  eomjjlex  qiuintitlcs  uivolrcs 
tiro  efjiuttiouH hctiueeii  red!  fjua/UUies,  fi)rnhed  by  cqaatiii^ 
the  nicinbcrs  of  real  and  Iniaginarij  units. 

Ilediictioii   of  Functions   of  i   to   the  Normal 

Form, 

338.    1.  If  wo  have  an  entire  function  of  /, 

a  +  ^i  +  ci^  4- f7i3  +  e/^ -f//5  +  etc., 
we  reduce  it  l)y  putting 

i^  —  —1,     i^  =  —  i,    i*  =  1,    etc.,     etc., 
and  the  expression  will  become 

(rt  —  c  +  e  —  etc.)  4- (5  —  fZ +/— etc.)  / ; 
■which,  when  we  put 

X  =  a  —  c  -\-  e  —  Qtc,        y  =  h  —  d  -\-f  —  Qic.j 
becomes  x  +  yi,  as  required. 

2.  To  reduce  a  rational  fraction  of  i  to  the  normal  form, 
wc  reduce  both  numerator  and  denominator.  The  fraction 
■will  then  take  the  form 

a  -I-  hi 

■ • 

m  +  ni 
Since  this  is  to  be  reduced  to  the  form  x  +  yi,  let  us  put 

a  +  hi 

— 7—.  =  X  +  yi, 

m  -\-  HI  -'  ' 

X  and  y  being  indeterminate  coefficients. 

Clearing  of  fractions, 

a  +  hi  =  mx  —  1UJ  +  {my  +  nx)  i. 


) 


IMA OINAR  7   Q  UANTITIES. 


397 


10  equation 


>.     Ilcnce, 

s  in  vol  res 
'  equatiiig 


Comparing  the  number  of  real  and  imaginary  units  on 
each  side  of  the  C(|uation,  we  have  tiie  two  equations 

mx  —  ny  =  a,  nx  +  my  =  b. 

Solving  them,  we  find 


X  = 


ma  +  nb 
m^'ii^ ' 


y  = 


mb  —  na 


Therefore, 


a  A-  bi  _  ma  +  nb   ,   mb  —  na 

111^  +  n* 


m  +  )ii        ?n'^  4-  n^ 
which  is  the  normal  form. 


'i  "' 


JVorinal 


'tc, 

nial  form, 
Q  fraction 


3t  us  put 


1^ 


EXERCISES, 

Reduce  to  the  normal  form  : 


I. 

2. 


1  +  i  —  v>  +  i^—  i»  —  i^+  i\ 
6  +  5i  1  +  i 


1-i 


1  —  * 


3- 
6. 


8. 


a  +  bi 


2 

i  —  1 

mi{x  —  ai) 

X  -\-  ai 
{a  +  bi){a— 

hi) 

{x  +  biY 


2  +  U  "'     a  —  hi 

10.  What  is  the  value  of  the  exponential  series  which  gives 
the  development  of  c»?    We  put  x  —  i  in  §  310,  Eq.  10. 

11.  Develop  (1  +  xiy  by  the  binomial  theorem. 

12.  What  are  the  developed  values  of 

(1  +  hiY  +  (1  -  ^'0'' 
and  (I  +  hiy^  —  (1  —  Z*/)«? 

13.  Write  eight  terms  of  the  geometrical  progression  of 
which  the  first  term  is  a  and  the  common  ratio  /. 

14.  Find  the  limit  of  the  sum  of  the  geometrical  progres- 
sion of  which  the  first  term  is  a  and  the  common  ratio  ^^• 

3*^9.  To  reduce  the  square  root  of  an  imaginary  expres- 
sion to  the  normal  form. 


Let  the  square  root  be  ^/a  +  bi. 

W^e  put  X  -{-yi  =  ^a  +  Vu 

Squaring,      7?  —  y^-\-  ^xyi  =  «  +  bi. 


It 


Hi 


=3«%Mn>t*>iw«* 


398 


IMA  GIN  A  nr  Q  UANTITIES. 


M 


Comparing  iniits,     7?  —  y^  =  a, 

2.1!/  =  b. 
Solving  this  pair  of  quadratic!  C(|uations,  wc  find 


X 


y 


Therefore, 


Vrt  +  l^i 


-V( 


VCVr 

5^  +  ^  + 

^) 

V2 

V{V'( 

?2  4-  />^  _ 

^0 

^/'Z 

v< 

1^  +  /y2  . 

f  rA  , 

4  / 

V</^  +  Z<2  _ 


'Z 


') 


EXERCISES. 

•Ecducc  the  square  roots  of  the  following  expressions  to  tlie 
normal  fonn: 

I.     3  +  U.  2.     4  +  3/.  3.     12  +  5t. 

4.  Find  the   square  roots  of  the  imaginary  unit  i,  and 

of  —  i,  and  prove  the  results  by  squaring  them. 

Note  that  this  comes  under  the  preceding  fomi  when  a  =  0  and 
6  =  ±1. 

5.  Find  the  fourth  roots  of  the  same  quantities  by  extract- 
ing the  square  roots  of  these  roots. 

J?30.  Quadratic  Eqnations  loUh  Imaginary  Roots.  The 
combination  of  the  preceding  operations  will  enable  us  to  solve 
any  quadratic  equation,  whether  it  does  or  docs  not  contain 
imaginary  quantities. 

Example  i.    Find  x  from  the  equation 

x^  4-  4a-  +  13  =  0. 

Completing  the  square  and  proceeding  as  usual,  we  find 
a;'-'  +  4a:  +  4  =  —  9, 

whence  a;  +  2  =  ^/—  9  =  ±  3/, 

and  X  =  —  2  ±  3i. 

Ex.  2.  7?  -\-  bxi  —  c  =  0. 

Completing  the  square, 

JT-  4-  o.n  —  —  =1  e  —  —' 
4  4 


— )•• 

lis  to  tlie 

5 1. 

t  /,  and 

I  =  0  and 
extract- 

^  The 
to  solve 
contain 


I'uid 


nfAGLYAIi  V   Q UANTITIEa. 
Extracting  the  root, 


309 


hi 


Vic  —  /;3 


whence 


EXERCISES. 

Solve  the  fjiiatlratic  cfiualioiis: 

I.     sfi -j- z -\-  I  =  0.  2.     a^^  — a:+l— 0. 

3.     .r*  +  3.<:  -f  10  =  0.  4.     x^  -f-  lOar  +  34  =  0. 

Form  quadratic  cqnations  (§  10!))  of  which  the  roots  shall  Ijc 
5.     a  +  bi  and  a  —  bi.  6.     ai  -f  b  and  ai  —  b. 

331.  Ejrpnnential  Fundionf^.  Wlien  in  the  exponential 
fiinotiun  a'  we  suppose  z  to  represent  an  imaginary  expre»iou 
X  -\-  iji,  it  hecomes 

Tins  expression  could  have  no  meaning  in  any  of  onr  pre- 
vious definitions  of  an  exponent,  because  we  have  not  shown 
what  an  imaginary  exponent  could  mean.  But  if  we  sup]xi£c 
the  etlect  of  the  exponent  to  be  defined  by  the  ex|)onfntial 
theorem  (§§  301^  314),  we  can  develop  the  above  expre&aiuQ. 
First  we  have,  by  the  fundamental  law  of  exponents, 

Next,  if  we  put  c  =.  Nap.  log  a,  we  have 

whence,  av^  =  e<^'. 

If  we  put,  for  brevity,  cy  =  u,  we  shall  now  have 

fix+yi  —  QCfQUi^ 

The  value  of  a^  being  already  perfectly  understo«xl,  we 
may  leave  it  out  of  consideration  for  the  present,  and  investi- 
gate the  development  of  e«*.  By  the  exponential  theorem 
(§  310,  10), 

e"*  =  1  +  id  -\- 


u^i^ 


+  -3-r  + 


IT 


2/5/5 


+  -5  r  +  <^tc. 


1! 


M! 


'I 


400  IVAfUNMlY  QUM^TITJJ'JS. 

Substituting  for  tlie  powers  of  i  their  values  (^  335), 

c«*  =  1  -  ^,  -{-  ^j  -  ,-j  +  etc.  4-  (.  -  3 J  +  5j  -  etc.)  ^ 

These  two  series  are  each  functions  of  it,  to  which  special 
names  iiave  been  given,  namely : 


n' 


W" 


?/• 


?r 


Ih'f.    Tiio  series    1  —  .r?  +  Tt  ~  m  +  ui  ~  ^^c.,  is  called 
the  cosine  of  //,  and  in  written  cos  ?^ 


?/"       ?<'       ?/'       ?<^ 
/><;/'.     The  tieries  n  —  .n  +  r  i  ~~  r^i  +  rr»  ""■  ^^^'^   ^"^  called 

the  sine  of  Uy  and  is  written  sin  u. 

Using  this  notation,  the  above  development  becomes, 

c«'  =  cos  ti  +  /  sin  Uf  (a) 

which  is  a  fundamental  ecjuation  of  Algebra,  and  should  bo 
memorized. 

Kkmarks.  These  functions,  cos  u  and  sin  n,  have  an  ox- 
tensive  use  in  both  Trigonometry  and  Algebra.  To  familiarizo 
himself  with  them,  it  will  be  well  for  the  student  to  compute 
their  values  from  the  above  series  for  i  =  0.25,  i  =  0.50, 
i  z=  1,  i  =z  2,  to  three  or  four  places  of  decimals.  This  can 
be  done  by  a  process  simil^K*  to  that  employed  in  computing  e 
in  §  310.    If  the  w'ork  is  done  correctly,  he  will  tind: 


For    u  = 


4' 


(( 


a 


(< 


n  = 


u  =  1, 
n  =  2, 


cos  7  =:       0.9G9, 
4 

cos  ,  —       0.878, 
2 

cos  1  =       0.540, 
cos  2  =  —  0.4in, 


sin  -r  ^  0.2-47. 
4 


sin  -  =  0.470. 

sin  1  =  0.841. 
sin  2  =  0.909. 


Iili2,  Let  us  now  investigate  the  properties  of  the  functions 
cos  u  and  sin  it,  which  are  detined  by  the  e([uatiuns, 


cos  u 


1  — 


u 


IC 


fr 


i!'^4!       0!  +  ^*^' 


11 


n^ 


w 


(») 


sin  «  =  «_-  +  -!_  ^-,  +  etc 


\ 


!2r.), 


lic'li  tipCL'iiil 
.,  is  called 

is  calJL'J 


slioiikl  1)0 


avc  an  cx- 
amiliarizo 
L)  coinpiito 
/  =  0.50, 
riiis  can 
u])uting  e 


|).2-i7. 

.470. 

.841. 
,900. 

riiiictions 


(i) 


j 


JMA  ajyj  H  Y   QUA NTITIES. 


401 


Since  cos  u  includos  only  even  powers  of  v,  its  value  will 
remain  iiiu-han^^ed  when  we  change  the  sign  of  u  I'runi  -f  to 
— ,  or  vice  I'tma.     Hence, 

cos  {—  n)  =  cos  u.  (1) 

Since  sin  it  contains  only  odd  powers  of  u,  its  sign  will 
ciiauge  with  that  of  u.     Hence, 

sin  (—  v)  =  —  sin  u.  (8) 

If  in  the  equation  («)  we  change  the  sign  of  //,  we  have, 
by  (1)  and  {'i), 

f>-iii  —  ^jos  (—  v)  +  i  sin  (—  //), 

or  c~"*  =  cos  u  —  /  sin  n. 

Now  multiply  this  e(|nati()n  by  (a).     Since 

1 


pUi 


X  e 


)-Ui   —    fUi 


=  e«»  X 


nUi 


=    1, 


we  have  1  =  (cos  uY  —  i^  (sin  vY, 

or  1  z=:  (cos  uf  +  (sin  iiy. 

It  is  customary  to  write  cos'  u  and  sin-  u  instead  of  (cos  v)' 
and  (sin  w)^,  to  ex})res8  the  S(|nare  of  the  cosine  ami  of  the 
sine  of  u.     The  last  equation  will  then  be  written 

cos'^  u  -}-  sin^  u  =  1.  {(•) 

Although  we  have  deduced  this  etiuaticm  with  entire  rigor, 
it  will  be  interesting  to  tost  it  by  S([uaring  the  equations  {b). 
First  squaring  cos  u,  we  lind  (§  2S-i), 

cos»  «  =  !-«»  +  n<  (i  +  ^  J  +  i)  -  etc. 

The  coefficient  of  ?i«  is  found  to  be 


+ 


+ 


n\  "^  2\{?i-  2)\  "^  4!  (M  -  4)!  "^  "^  w! 

when  n  is  doul)le  an  even  number,  and  to  the  negative  of  this 
exi)rest;ion  when  u  is  doulde  an  odd  number. 
Again,  taking  the  square  of  sin  k,  wc  find 

siu'  u  =  «»  +  «'  (-  ^j,  -  j-ji  j)  +  etc. 
30 


402 


LVA  GINAR  Y  Q  UANTITIES. 


the  coefficient  of  m"  bein": 


1!  {n  —  l)\      3!  {)i  —  'd)\      5!  {n  —  b)\ 


•    •    •    t 


{n-\)\  1!' 

or  the  negative  of  this  expression,  according  as  -  n  is  even  or 
odd. 

Adding  sin'^  u  and  cos^  u,  we  sec  that  tlic  terms  u^  cancel 
each  other,  and  that  tlie  sum  of  the  coefficients  of  u^  can  be 
arranged  in  the  form 

4!      1!  3!  "^2!  2!      3!  l!  "^4!* 

Let  ns  call  this  sum  A.  If  we  multijjly  all  the  terms  by 
4 ! ,  and  note  that  by  tlie  general  form  of  the  binomial  coeffi- 
cients, 

n\         _  hi\ 

s\  {n-s)\  ~  \J' 

which  sum  is  zero,  by  §  20:^,  Th.  11.    Therefore  the  coefficients 
of  «"  cancel  each  other. 

Taking  the  sum  of  the  coefficients  of  u'",  we  arrange  them 
in  the  form 


+ 


+  etc., 


^1 


4 


?i\       l\{n  —  l)\^  2l{H  —  2)l       3l{n-l])\ 
which  call  A.     Tlien  multiplying  by  n\,  we  liave 

'"^=^-(;va-(a)+----+(;:)' 

which  sum  is  zero.  Therefore  all  the  coefficients  of  u'^  cancel 
each  other  in  the  sum  sin^  n  -\-  cos^  n,  leaving  only  the  first 
term  1  in  cos^  v,  thus  ])roving  the  equation  (r)  indejiendeutly. 
This  exam})le  illustrates  the  consistency  which  pervades  all 
branches  of  mathematics  when  the  reasoning  is  correct.  The 
conclusion  (r;)  was  reached  by  a  very  long  process,  resting  on 
many  of  the  fundamental  i)rinciples  of  Algebra  ;  and  on  reach- 


IMA GINAR  T  Q  UANTITIE8, 


403 


I 


n-l)\lV 
*  is  even  or 

tis  u^  cancel 
f  w^  can  be 


lie  terms  by 
jmial  cocfli- 


coefiicients 


^ange  tlicm 


+  etc., 


nr 


I  u^  cancel 
ily  the  first 
pendently. 
>ervades  all 
rect.  The 
resting  on 
1  on  reach- 


ing a  simple  conclusion  of  this  kind  in  such  a  way,  the  mathe- 
matician always  likes  to  test  its  correctness  by  a  direct  i)rocess, 
when  possible. 

Let  us  now  resume  the  fundamental  equation  {a).  Since 
u  may  here  be  any  quantity  whatever,  let  us  put  mi  for  u. 
The  equation  then  becomes. 

But  by  raising  the  equation  {a)  to  the  n*^  power,  we  have 


nnui 


(cos  11  -I-  i  sin  iiY. 


Hence  we  have  the  remarkable  relation, 

(cos  u  4-  i  sin  ?<)"■  =  cos  nu  -f  i  sin  nu. 
Supposing  ?^  =  2,   and  developing  the  first  member,  we 


have 

cos^  u  —  %\v?  u  -{-2i  sin  u  cos  u  =  cos  2u  -\-  i  sin  2rt. 
Equating  the  real  and  imaginary  parts  (§  327,  Cor.),  we  have 
cos^  u  —  sin^  u  =  cos  2i(, 
2  sin  u  cos  ti  =  sin  22(, 

relations  which  can  be  verified  from  the  series  representing 
cos  u  and  sin  u,  in  a  way  similar  to  that  by  Avhich  we  verified 
sin^  it  -f  cos^  ic  =  1. 

EXERCISES. 

1.  Find  the  values  of  cos^  u,  sin^  w,  cos*  u,  and  sin'*  u  by 
the  preceding  process. 

2.  AVrite  the  three  equations  which  we  obtain  by  putting 
11  =  a,  u  =z  b,  and  u  =z  a  +  b  in  equation  {a).  Tlien  equate 
the  product  of  the  first  two  to  the  third,  and  show  that 

cos  {n  ■\-  b)  =.  cos  a  cos  b  —  sin  a  sin  b, 
sin  {a  +  b)  ■=.  sin  a  cos  J  -f-  cosw  sin  b. 

3.  Reduce  to  the  normal  form, 

{x  —  i)  {x  —  2i)  {x  —  30  {x  --  U). 

4.  Develop  {a  +  bi)^  by  the  binomial  theorem,  and  reduce 
the  result  to  the  normal  form. 


,  )■ 


1  ( ! 


404 


GEOMETRIC  REP  RESENT  A  TION. 


CHAPTER     II. 


'I 


THE    GEOMETRIC 


REPRESENTATION 
QUANTITIES. 


OF    IMAGINARY 


333.  In  Algebra  and  allied  branches  of  the  higher  mathe- 
matics, the  fundamental  operations  of  Arithmetic  are  extended 
and  generalized.  In  Elementary  Algebra  we  have  already  had 
several  instances  of  this  extension,  and  as  we  are  noAv  to  have 
a  much  wider  extension  of  the  operations  of  addition  and  mul- 
tiplication, attention  should  be  directed  to  the  principles 
involved. 

In  the  beginning  of  Algebra,  we  have  seen  the  operation  of 
addition,  which  in  Arithmetic  necessarily  implies  increase^  so 
used  as  to  produce  diminution. 

The  reason  of  this  is  that  Arithmetic  does  not  recognize 
negative  quantities  as  Algebra  does,  and  therefore  in  employ- 
ing the  latter  we  have  to  extend  the  meaning  of  addition,  so  as 
to  apply  it  to  negative  quantities.  Wlien  thus  applied,  we 
have  seen  that  it  should  mean  to  subtract  the  quantity  which 
is  negative. 

In  its  primitive  sense,  as  used  in  the  third  operation  of 
Aritlimetic,  the  word  multiphj  means  to  add  a  quantity  to  itself 
a  certain  number  of  times.  In  this  sense,  there  would  be  no 
moaning  to  the  words  "multiply  by  a  fraction."  But  we  ex- 
tend the  meaning  of  the  word  multiply  to  this  case  by  defining 
it  to  mean  taking  a  fraction  of  tlie  quantity  to  be  multiplied. 
We  then  find  that  the  rules  of  multiplication  will  all  apply  to 
this  extended  operation. 

This  extension  of  multiplication  to  fractions  docs  not  take 
account  of  negative  multipliers.  In  the  latter  case  Ave  can 
extend  the  meaning  of  the  operation  by  providing  that  the 
algebraic  sign  of  the  quantity  shall  be  changed  when  the  mul- 
tiplier is  negative.  We  thus  have  a  result  for  multiplication 
by  every  positive  or  negative  algebraic  number. 

Now  that  wc  have  to  use  imaginary  quantities  as  multi- 


i(?  " 


if 


IMAGINARY 


higher  mathe- 
c  are  extended 
v^e  already  had 
'e  now  to  have 
ition  and  mul- 
the  princii^les 

e  operation  of 
2S  increase,  so 

not  recognize 
)re  in  emploj- 
iddition,  so  as 
s  applied,  wo 
Liantity  which 

operation  of 
mtity  to  itself 

would  be  no 

But  we  ex- 

3e  by  defining 

)e  multiplied. 

1  all  apply  to 

does  not  take 
case  Ave  can 
ing  that  the 
len  tlie  mul- 
lultiplication 

es  as  multi- 


GEO METRIC   REPliESENTA  TION. 


405 


pliers,  a  still  further  extension  is  necessary.  ITifherto  our 
operations  with  imaginary  units  have  been  purely  symbolic; 
that  is,  we  have  used  our  symbols  and  i)erl'ormed  our  o])erati(>us 
without  assigning  any  detinite  meaning  to  them.  We  shall 
now  assign  a  geometric  signification  to  operations  with  inuigi- 
nary  units,  subject  to  these  three  necessary  conditions  : 

1.  The  operations  must  be  subject  to  the  same  rules  as 
those  of  real  quantities. 

2.  The  result  of  operating  with  an  imaginary  quantity 
must  be  totally  different  from  that  of  operating  with  a  real  one, 
and  llic  imaginary  quantity  must  signify  something  which  a 
real  quantity  does  not  take  account  of. 

3.  If  the  imaginary  quantity  changes  into  a  real  one,  tlio 
operation  must  change  into  the  corresi)onding  one  with  real 
quantities. 

334.  Geometric  Iicpresentation  of  Tinauinary  Units.  Cer- 
tain propositions  respecting  the  geometric  re})resentation  of 
multiplication  have  been  fully  elucidated  in  Part  I,  and  arc 
now  repeated,  to  introduce  the  corresponding  representations 
of  complex  quantities. 

I.  All  real  numbers,  positive  and  negative,  may  be  arranged 
along  a  line,  the  positive  numbers  increasing  in  one  direction, 
the  negative  ones  in  the  opposite  direction  from  a  fixed  zero 
point.  Any  number  may  then  be  represented  in  magnitude 
by  a  line  extending  from  0  to  the  place  it  occupies. 

We  call  this  line  a  Vector. 

II.  If  a  number  a  be  multiplied  by  a  positive  multiplier 
(for  simplicity,  suppose  +1),  the  direction  of  its  vector  will 
remain  unaltered.  If  it  be  multiplied  by  a  negative  multiplier 
(suppose  — 1),  its  vector  will  be  turned  in  the  opposite  dircc- 
tit)n  (from  0  —  «  to  0  +  «,  or  vice  versa).  Compare  §  73, 
where  the  coarse  lines  are  the  vectors  of  the  several  quantities. 


—  a 


+  a 


III.  If  the  number  be  multiplied  twice  by  —  1,  that  is,  by 
(—1)2,  its  vector  will  be  restored  to  its  first  position,  being 
twice  turned,  and  if  it  be  multiplied  twice  by  +  1,  that  is,  by 
(+  \y,  its  vector  will  not  be  changed  at  all.     Its  vector  will 


406 


IMA GINARY  Q UAyilTIES. 


'I 


+ia 


—  a 


+a 


— ta 


therefore  be  found  in  its  first  position,  whether  we  multiply  it 
by  the  square  of  a  j^.-^itive  or  of  a  negative  unit;  in  other 
words,  both  s(iuures  are  positive. 

IV.  To  multiply  the  line  +  a  twice  by  the  imaginary  unit 
i,  is  the  same  as  multii)lying  it  by  ir  or  —  1.     Hence, 

MiiUi])lijing  by  the  inia >Ji tiarij  unit  i  rmcst  ^ire  the 
vector  siicli'  a  niution  as,  if  repeated,  will  cJiange  it  froDv 
-\-  a  to  —  a. 

Such  a  motion  is  given  by  turn- 
ing the  vector  through  a  righ  t  angle, 
into  the  position  -f-  ia.  A  second 
motion  brings  it  to  the  position 
—  a,  the  opposite  of  -f-  «•  A  third 
motion  brings  it  to  —  iV/,  a  position 
the  oi)posite  of  +  ia.  A  fourth 
motion  restores  it  to  the  original 
po.'^ition  -}-  ii' 

If  we  call  each  of  these  motions  mvUiplyimj  oij  i,  we  have, 
from  the  diagram,  a  =  a,  ia  =  ia,  i\i  =  —  a,  i^a  =:  —  ia, 
i^a  =  a,  which  corresponds  exactly  to  the  law  governing  the 
powers  of  i  (§  325).     Hence  : 

//  a  quantity  is  represented  hy  a  vector  extending 
from  a  zero  point,  the  inuJtipliention  of  this  qunntity  hy 
the  imaginary  unit  may  he  represented  hy  turning  the 
vector  through  90°. 

V.  In  order  that  multiplier 
and  multiplicand  may  in  this  op- 
eration be  interchanged  without 
affecting  the  product,  we  must 
suppose  that  the  vertical  line 
which  we  have  called  ia  is  the 
same  as  ai,  that  is,  that  this  line 
represents  a  imaginary  units. 

We  have  therefore  to  count 
the  imaginary  units  along  a. 

vertical  line  on  the  same  system  that  we  count  the  real 
units  on  a  horizontal  line. 


—4  -3  -2  —1 


+  4t 
+  3i 
+  2i 


—I 

— 3i 
-4i 


we  multiply  it 
unit;  in  other 

imaginary  unit 
ieuee, 

must  ^ive  the 
hange  it  froDh 

+ia 


+a 


—m 


f)  bij  i,  we  have, 
a,  ihi  =  —  m, 
'  governing  the 


for  extending 
is  qunntitij  by 
y  turning  the 

— 

+  4i 

+  3i 

- 

+  2i 

I 

+  M      2      3      4 

-t 

- 

-2i 

- 

-3i 

_ 

— U 

count  the  real 


GEOMETRIC   REPliESENTA  TIOX 


407 


-a+bi 


bi 


-bi 


■a+bi 


U 


~a 


'a 


-a~bi 


N 


bi 

bi 


335.  Geometric  Eeprcsentation  of  a  Comjilex  Qnanlity. 
We  have  shown  (§  15)  that  a]gel)raic  addition  may  be  represented 
byjnitting  lines  end  to  end,  the 
zero  i)oint  ot*  eaeh  line  added  be- 
ing at  the  end  of  the  line  next 
preceding.  The  distance  of  the 
end  of  the  last  line  from  tlie  zero 
point  is  the  algebraic  sum. 

On  the  same  system,  to  repre- 
sent the  algebraic  sum  of  the  real 
and  imaginary  qnantities  a  -{■  bi, 
we  lay  otf  a  units  on  the  real  (horizontal)  lino,  and  then  b 
units  from  the  end  of  this  line  in  a  vertical  direction.  'J'he 
end  of  the  vertical  line  will  then  be  the  position  corresponding 
to  «  +  bi. 

It  is  evident  that  we  should  reach  the  same  point  if  wo 
first  laid  off  /•  '^aits  from  0  on  the  imaginary  line,  and  then  a 
unita  horizon  v  \      Ilence  this  system  gives 

bi  -\-  a  =  a  -{■  bij 

as  it  ought  to,  to  represent  addition. 

If  a  or  b  is  negative,  it  is  to  be  laid  off  in  the  opposite  di- 
rection from  the  positive  one.  We  then  have  the  points  cor- 
responding to  —  a  -\-  bi,  —  a  —  bi,  and  a  —  bi,  shown  in  the 
diagram,  which  should  be  carefully  studied  by  the  pupil. 

The  result  we  have  reached  is  the  following: 

Every  complex  quantity  a  +  bi  is  considered  as  be- 
lovging  to  a  certain  point  on  the  plane,  namely,  that 
point  which  is  reached  by  laying  off  from  the  zero  jwiiit 
a  units  in  the  horizontal  direction  and  b  units  in  the 
vertical  direction. 


330.  Addifion  of  Com- 
plex Qfuintities.  If  we  have 
several  complex  terms  to 
add,  as  a  -\-  bi,  m  —  ni, 
p  +  qi,  we  may  lay  them 
off  separately  in  their  ap- 
propriate magnitude  and  di- 


m 


qi 


\'\ 


n 


4U8 


LVA  OINA  liY  Q  UA  NTITIES. 


'\ 


reef  ion,    as  in   the   figure,   the   last  lino    terminating    i'     a 
jioint  R. 

If  we  first  add  tlie  quantities  a  +  hi,  etc.,  algebraically 
(§  32-1),  the  result  will  be 

a  +  m  +  ;;  +  {h  —  n  -^  q)  i. 

We  may  lay  of!f  this  sum  in  one  operation.  The  sum  a-\-m 
-\-])  Avill  carry  us  fi'oni  0  to  M,  and  the  sum  {b  —  n  -\-  q)  i 
from  M  to  IX,  because  MR  =  b  —  n  -\-  q.  Therefore  we  shall 
reach  the  same  jioint  R  whether  we  lay  the  quantities  off  sepa- 
rately, or  take  their  sum  and  lay  off  its  real  and  imaginary 
parts  separately. 

33*7.  Vectors  of  Complex  Quantities.  The  question  now 
arises  by  what  straight  line  or  vector  shall  we  represent  a  sum 
of  complex  quantities  ?     The  answer  is : 

Hie  vector  of  c;  suvi  of  sev- 
eral  vectors  is  the  straight  Una 
from  the  hcoinning  of  the  first 
to  the  end  of  the  last  vector 
added. 

For  example,  the  sum  of  the 
quantities  OX  =  a  and  XP  =  hi  is  the  vector  OP. 

It  might  seem  to  the  student  that  the  length  of  the  vector  represent- 
ing the  sum  should  be  equal  to  the  combined  lengths  of  all  the  separate 
vectors.  This  diificulty  is  of  the  same  kind  as  that  encountered  by  the 
beginner  in  finding  the  sum  of  a  positive  at\d  negative  quantity  less  than 
either  of  them.  The  solution  of  the  diificulty  is  simply  that  by  addition 
we  now  mean  something  different  from  both  arithmetical  and  algebraic 
addition.  But  the  operation  reduces  to  arithmetical  addition  when  the 
quantities  are  all  real  and  positive,  because  the  vectors  are  then  all  placed 
end  to  end  in  the  same  straight  line.  Therefore  there  is  no  inconsistency 
between  the  two  operations. 

Two  imaginary  quantities  are  not  equal,  unless  both  their 
real  and  imaginary  parts  arc  equal,  so  that  their  sum  shall  ter- 
minate at  the  same  i)oint  P.  Their  vectors  will  then  coincide 
with  each  other.     Hence : 

Tn'o  vectors  are  not  considered  equal  unless  they  agree 
in  direction  as  well  as  leni^th. 


i 


GEOMETRIC  REPRESENTA  TION. 


400 


natinff    i'     '^ 


ali^cbraicully 


'lie  sum  a  ■¥  m 
(^,  _  ^  +  q)  i 
•cforo  we  shsill 
itities  off  scpa- 
uul  imaginary 

question  now 
cprcsent  a  sura 


hi 


X 


or. 

vector  represent- 
of  all  the  separate 
iicountered  by  the 

uantity  less  than 
that  by  addition 

ical  and  algebraic 
addition  when  the 

are  then  all  placed 
is  no  inconsistency 

nless  both  their 
sum  shall  ter- 
1  then  coincide 


dess  they  a^ree 


In  other  worrls,  in  orilrr  in  drfrnnlnr-  a  vrrfnv  mn]- 
pJctt'hj,  we  must  know  i/s  direction  as  well  as  its  len^lli. 

This  result  embodies  the  theorem  of  the  i)recediiig  chapter  (j;  o^i  t, 
that  two  complex  (|uantities  are  not  e<jiial  unless  both  their  reiil  ai:d 
imafrinarj'  i)arts  are  e()ual.  It  is  only  in  ciise  of  this  double  ecjuality  that 
the  two  complex  quantities  will  belong  to  the  same  point  on  the  i)lane. 

Because  OXP  is  a  right  angle,  we  have  by  the  Pythagorean 
theorem  of  Geometry, 

(length  of  vector)^  =.  a^  -{■  h\ 


or 


I 


length  of  vector  =  V«^  +  h\ 

We  are  careful  to  say  length  of  vector,  and  not  rr.rely  vec- 
tor, because  the  vector  has  dircctio7i  as  well  as  length,  and  the 
direction  is  as  important  an  element  as  length. 

To  avoid  repeating  the  words  ''  length  of,"  we  shall  put  ai» 
dash  over  the  letters  representing  a  vector  when  we  consider 
only  its  length.     Then  OX  will  mean  length  of  the  line  OX. 

Def.  The  length  of  the  vector,  or  the  expression 
Vrt^  +  t)\  is  called  the  Modulus  of  the  comi)lex  ex- 
pression a  +  bi. 

The  modulus  is  the  absolute  value  of  the  expression,  con- 
sidered without  respect  to  its  being  positive  or  negative,  real 
or  imaginary.     Thus  the  different  expressions, 

—  5,     +5,     3  +  4/,     4  —  3i,    5/, 

all  have  the  modulus  5  (because  V^^  _|_  42  —  5).  The  points 
which  represent  them  are  all  5  units  distant  from  the  zero 
point,  and  so  lie  on  a  circle,  and  their  vectors' are  all  5  units  iu 
length. 

The  German  mathematicians  therefore  call  the  modulus 
the  absolute  vcdue  of  the  complex  quantity,  and  this  is  really 
a  'setter  term  than  the  English  expression  modulus. 

Def.  Tlie  Angle  of  the  vector  is  the  angle  which  it 
makes  with  the  line  along  which  the  real  units  are 
measured. 

If  OA  is  this  line,  and  OB  the  vector,  the  angle  is  AOB. 


"W 


410 


IMAGINARY  QUANTITIES. 


EXERCISES, 


M 


Liiy  off  the  following  complex  qnantiHc?,  draw  Hio  vectors 
correspoiuling  to  thciu,  imd  iind  the  modulus  both  by  measure- 
ment and  calculation  : 


I. 

4- 

7. 
10. 

13. 
16. 


2. 

4  -  'M. 

3. 

—  4  +  3/. 

5^ 

3  +  4/. 

6. 

3  —  4t. 

8. 

—  3  -  4/. 

9. 

5  +  7/. 

II. 

5  +  5/. 

12. 

5  +  4t. 

14. 

3  +  i. 

15. 

3  —  L 

4  +  3/. 

-  4  -  3/. 

—  3  +  4i. 

5  +  0/. 
3  +  2*. 
3  -  2i. 

17.  Draw  a  horizontal  and  vertical  line;  mark  several 
points  on  the  plane  of  thei<e  lines,  and  find  by  measurement 
the  complex  expressions  for  each  point.  Also,  draw  the  sev- 
eral vectors  and  measure  their  length.  Continnc  this  exercise 
until  the  relation  between  the  complex  expressions  and  their 
points  is  well  apprehended. 

Note.  The  student  may  adopt  any  scale  he  pleases,  but  a 
scale  of  millimeters  will  be  found  convenient. 

338.  Geometric  MuUiph'cafion.  The  question  next  arises 
whether  the  results  we  obtain  for  multiplication  of  complex 
quantities  follow,  in  all  respects,  the  usual  laws  of  multiplica- 
tion, especially  the  commutative  and  distributive  laws. 

I.  To  imdtiply  a  vector  hy  a  real  factor. 

Let  the  vector  be  «  -}-  hi  and  the 
factor  m.     The  product  will  be  ^ 

ma  4-  mli. 

In  the  geometric  construction,  let 
OA  r=  «  and  AB  =  Z>i.  We  shall 
then  have,  by  the  rule  of  addition, 

Vector  OB  —  a  -\-  hi. 

"When  we  multiply  a  hjm,  let  OA'  be  the  product  ma,  and 

A'B'  the  product  mhi.     Because  the  lines  OA  and  AB  are  both 

multiplied  by  the  same  real  factor  in  to  form  OA'  and  A'B',  wo 

shall  have 

OA  :  AB  :  OB  =  OA'  :  A'B'  :  OB'. 


^•1 


aw  Hic  vectors 
til  by  mcjisiire- 

-  4  +  3i. 
;  —  4i. 
)  +  7i. 
\  +  4i. 
I  -  i. 

mark  several 
'  iiicasuremciit 

draw  the  sov- 
le  this  exercise 
ions  and  tlieir 

pleases,  but  a 

on  next  arises 
n  of  complex 
of  multiplica- 
laws. 


duct  ma,  and 

AB  are  both 

and  A'B',  we 


I    : 


GEO  METRIC   REP  RESENT  A  TION. 


411 


Therefore  the  triangles  OAB  and  OA'B'  are  similar  and 
Of|uiaugular,  so  that 

angle  AOB'  =  angle  AOB. 

This  shows  that  the  lines  OB  and  OB'  coincide,  so  tJiat 
BB'  is  the  continuation  of  OB  in  the  same  straight  line.  More- 
over, the  above  proportion  gives 

OB'  =  wOB, 
or,  from  (1),  vector  OB'  =  m  vector  OB. 

Therefore,  DiuUiplijing  a  vector  hy  a  rrrd  factor 
changes  its  lcu!:!th  without  altering  its  direction. 

II.  To  multiply  a  vector  hij  the  iimiglnanj  unit. 

Multiplying   a  +  hi  by  /,  the        _^q 
result  is 

—  b-{-  ai. 

The  construction  of  the  two 
vectors  being  made  as  in  the  fig- 
ure, we  have 

OB  =  «  +  hi, 
Oq  =  -h  +  ai. 

Because  the  triangles  OPQ  and  OAB  are  right-angled  at  P 
and  B,  and  have  the  sides  containing  the  right  angle  e(pud  in 
length,  they  are  identically  ecpial,  and 

angle  POQ  =  angle  OBA  =  90°  -  angle  BOA. 

Hence  the  sum  of  the  angles  POQ  and  BOA  is  a  right 
angle,  and  because  POxV  is  a  straight  line,  therefore, 

angle  BOQ  =  90°. 

Therefore,  the  result  of  multiplying  the  vector  OB  hij 

the  iinaginary  unit  is  to  turn  it  90°  without  cJuuiginjJ 

its  length. 

We  have  assumed  this  to  be  the  case  when  the  vector  represents  a 
real  quantity,  or  lies  along  the  line  OB  ;  we  now  see  that  the  same  tiling 
holds  true  when  the  vector  represents  a  complex  quantity. 

If  instead  of  the  multiplier  being  simply  the  imaginary 
unit,  it  is  of  the  form  ni,  then,  by  (I),  in  addition  to  turning 
the  vector  through  90°,  we  multiply  it  by  n. 


412 


IMAOINAltY  qUANTITIES. 


'» 


III.  To  multiply  a  vector  hij  a  complex  quantity, 

m  -f  ni, 

Tliis  will  consist  in  niiiltii)Iying  sopanitely  l»y  m  and  i/i, 
and  'ulding  the  two  products.     Put  OB  =  a  -\-  Oi,  the  vector 
to  be  multiplied  ;  ON  = 
i/i  +  ni,  the  multiplier. 

To  multiply  OIJ  by  m, 
wo  take  a  length  OC,  deter- 
mined by  the  proportion, 

0C:0B  =  7n:  1,    (I) 

whence  by  (I), 

00  =  w-OB 
=  m  {a  +  bi). 

To  multiply  OB  by  ni,  wo  take  a  length  CD  determined 

by  the  condition, 

length  CT)  =  n  length  OB, 


or  CD  :  OB  =  w  :  1 ; 

and  to  multiply  by  i,  wc  place  it  peiTJendicnlar  to  OB.      (II) 

AVe  then  have, 

CD  =:  OB  X  ni. 

In  order  to  add  it  to  OC,  the  other  product,  we  place  it  as 

in  the  diagram,  and  thus  find  a  point  D  which  corresponds  to 

the  sum 

OC  +  CD  =  0Bxm-\-0Bx7ii; 

that  is,  to  the  product 

(m  -f  7ii)  {a  +  bi). 

Now  because  OC  =  OB  x  m  and  CD  =  OB  x  n,  we  haye 


(jG   :  CD  =m:7i  =  0M  :  MN, 

and  because  the  angles  at  M  and  C  are  right  angles,  the  tri- 
angles OCD  and  OMN  are  similar.     Therefore, 

angle  COD  =  angle  MON. 

Ilence  the  angle  AOD  of  the  product-vector  is  equal  to  the 
sum  of  the  angles  of  the  multiplier  and  multiplicand. 
For  the  length  OD  of  the  product-vector  we  have, 


I 


antity, 

l»y  m  iiiul  II  i, 
1)1,  tliu  vector 


D  determined 


to  OB.      (II) 


we  place  it  as 
corresponds  to 


}  X  n,  we  have 
angles,  the  tri- 


is  eqnal  to  tlio 

icand. 

!  have, 


GEOMETRIC  REPRESENTA  TION. 

length  Ob^  =  OU^  +  CD* 

=  m^OB^  +  )iH)]? 

Extracting  the  square  root, 

length  OD  =      Vin'^  +  n^  •  OB 


413 


Therefore  the  length  of  the  product-vector  is  equal  to  the 
products  of  the  lengths  of  the  vectors  of  tlie  factors. 

Combining  these  t\vo  results,  we  reach  the  conclusion: 

77ic  modulus  of  tJie  product  of  two  complex  factors  is 
cqu(d  to  the  product  of  tlieir  moduli. 

Tlie  angle  of  tJie  product  is  equal  to  the  suni  of  the 
angles  of  tJic  factors. 

SSO.  Tlie  Roofs  of  Unify.  Wo 
have  the  following  curious  pr()l)lem: 

Given,  a  vector  OA,  which  call  a; 
it  is  required  to  find  a  complex  factor 
X,  such  that  when  we  multi})ly  a  n 
times  by  x,  the  last  product  shall  be  a 
itself.     That  is,  we  must  have 

x^a  --  a. 

The  required  factor  must  be  one 
which  will  turn  the  vector  round  without  changing  its  length. 
Let  us  begin  with  the  case  of  n  — :  3. 

Since  three  equal  motions  must  restore  OA  to  ics  original 
position,  the  condition  will  be  satisfied  by  letting  x  indicate  a 
motion  tlirougli  120'',  so  that  OA  shall  take  the  position  OB 
wiien  angle  AOB  =  120°.  Then,  P  being  the  foot  of  the  jier- 
pondicular  from  B  upon  AO  oroduced,  we  shall  have  angle 
FOB  =  60°,  and  angle  PBO  =  30^     Therefore, 


PO  =  j«, 


m  =  f  . 


and 


vector  OB  =.  xa  =.  —  ,,a  -I -ai. 


•  ' 


414 


/.»M (UNA ItY  QUA NTITIES. 


*l 


lU'canso  llio  factor  ir  Iwis  not  cliimffcd  tlio  lon^'tli  of  tlic  line, 
the  inodulus  of  x  in  unitv,  and  lu'caiiso  it  lias  turned  the  lino 
tlirougli  l'iO°,  its  angle  is  120°.     Therefore  its  vuluu  is 

-  OP  +  PIU" 

on  a  scale  of  numbers  in  which  OB  =:;  1 ;  that  is, 

1    ,    \/3. 

Reaponing  in  the  same  way  with  respect  to  the  i)r()duct ;?%, 
"which  produces  the  vector  OC,  "we  llnd 


x^  = 


•> 


V3. 


an  equation  which  wc  readily  prove  by  squaring  the  preceding 
value  of  X  and  reducing. 

Multiplying  these  values  of  ;r  and  x^,  we  find 

x^  ==  1, 

which  ought  to  bo  the  case,  because  x^a  =  a.     Ilencc, 

1      a/'J 
TJie  complex  quantity  —  ,j  +  ~j-i  is  a  cube  root  of 

unity. 

But  the  vector  OC,  of  which  the  angle  is  240°,  also  repre- 
sents a  cube  root  of  unity,  if  we  suppose  OC  :=  1,  becaune 
three  motions  of  240°  each  turn  a  vector  through  720°,  or  two 
revolutions,  and  thus  restore  it  to  its  original  position.  This 
also  agrees  with  tlie  algebraic  process,  because,  by  squaring  the 
above  value  of  x%  we  have 

/     1       V3Y_1       3       V'3._       1   ,   V3._ 
\     2~    2   7-4~4+2*-~2+2*-^' 

and  by  repeating  the  process  we  find 

/    1     a/3.\/    1     VaA     / 

Since  1  itself  is  a  cube  root  of  unity,  because  1^  =  1,  we 
conclude  : 

TJicre  arc  three  cube  roots  of  unity. 


OKOMETIUV  UEPltHSHSTA  TION. 


415 


(1)  of  tlic  liiu', 
inuil  the  line 
uluu  id 


ic  product  .f'^a, 


the  prcccclin 


Hon  CO, 
ciibe  root  of 

0°,  also  rcprc- 
=  1,  bectiuso 
1  7;>0°,  or  two 
)ositi()n.  This 
ly  squaring  the 


We  rcudily  Und,  l)y  the  process  of  >>  'S.W,  IV,  that 

/',     —  1,     —  /,     and     1, 

arc  all  fourth  roots  of  unify. 

By  a  course  of  reasoning  similar  to  the  above  for  any  value 
of  n,  we  conclude  : 

Tlia  «"•  roots  of  unity  arc  n  in  numhcr. 

EXERCISES. 

1.  Form  the  first  eight  })owers  of  the  expression 

1     ,_!_.. 

sliow  that  the  eighth  power  is  1,  and  lay  off  the  vector  corre- 
sponding to  each  power. 

2.  Form  the  first  twelve  powers  of 

V3       1  . 

and  show  that,  the  twelfth  power  is  +1. 

3.  Find  the  fifth  and  sixth  roots  of  unity  by  dividing  the  cir- 
cle into  five  and  six  parts,  and  either  computing  or  measuring 
the  lengths  of  the  lines  which  determine  the  expression. 

Note.  The  student  will  remark  the  similarity  of  the  gen- 
eral problem  of  the  n^^^  roots  of  unity  to  that  of  dividing  the 
circle  into  n  equal  parts  (Geom.,  Book  VI). 


r    ) 


\/3. 


ise  18  =  1,  we 


BOOK    XIII. 

THE    GENERAL     THEORY    OF  EQUA- 
TIONS, 


'I 


Vi  = 
etc. 


Every  Equatioi*  has  a  Root. 

3-40.  In  Booi.  Ill,  equations  containing  one  unknown 
quantity  were  reduced  to  the  normal  form 

Aaf'  +  Bx^-^  +  Cx^-^  + +  J^  =  0. 

If  wc  divide  al]  the  terms  of  this  equation  by  the  coefficient 
A,  and  put,  for  br/vity, 

B 

a     ' 

A' 

etc. 
F 

the  equation  win  '  cPomo 

a;»  +  p^x'^--^  4- ;  5,:^'^""''  +  .  .  .  .  +  pn-i^  +  pn  =  0.       (a) 

This  equation  is  cailod  the  General  Equation  of  the 
^j,''*  Degree,  because  it  is  the  form  to  wliicli  every  algebraic 
cfiuation  can  be  reduced  by  assigning  the  proper  values  to  w, 
and  to  ^1,  p^,  ;>3,  etc. 

Tlie  71  quantities  Pi,  P2,  .  .  .  -  Pn  ^yc  called  the  Coeffi- 
cients of  the  equation. 

We  may  consider  pn  .is  the  coefficient  of  .^*^  =  1. 

3H .  Thkorf.m  I.  Every  (fjjjcbt'aic  cqiuitioii  has  a  root, 
real  or  iinaghiary. 

That  is,  whatever  numbers  we  may  put  in  place  of  .7?,,  p^, 
p^,  .  .  .  .  pn,  there  is  always  some  oxpres>^ion,  real  or  imaginary, 
which,  being  substituted  for  x  in  the  equation,  will  satisfy  it. 


GENERAL    TUSORT   OF  EQUATIONS. 


417 


F  EOUA- 


t. 

one   unknown 

=  0. 

the  coefficient 


Vi  =  0.      {a) 

ation  of  the 

very  algebraic 
er  values  to  Hf 

the  Coeffi- 

rl. 

on  has  a  root, 

ace  o£.Pi,Pz, 
or  imaginary, 
ill  sutiisiy  it. 


Rem.  The  theorem  that  every  equation  has  a  root  is  demonstrated  in 
special  treatises  on  the  theory  of  equations,  but  the  demonstration  is  too 
long  to  be  inserted  ht;re. 

If  we  suppose  the  values  of  the  coefficients  PdPq,  etc.,  to 
vary,  the  roots  will  vary  also.     Hence, 

TiiEOKEM  II,     TJbG  roots  of  ail  algebraic  equation  ai  'j  • 
fiiiictlojis  of  its  eoefficiciits. 

Example.     In  Chapter  VI  we  have  shown  that  the  roots 

of  a  (piacL'atic equation  are  functi(jns  of  the  coefficients,  because 

if  the  equation  is 

x^  -\-  px  -i-  q  =  0, 


the  root  is 


X 


—  P±  Vf—q 


'Z 


which  is  a  function  of  ^  and  q. 


342.  Equations  ivliich  caji  he  solved.  If  the  degree  of  the 
equation  is  not  higher  than  the  fourtli,  it  is  always  possil)Ie  to 
express  the  root  algebraically  as  a  function  of  the  coefficients. 

But  if  the  equation  is  of  the  fifth  or  any  higher  degree,  if; 
is  not  possible  to  express  the  value  of  the  root  of  the  general 
equation  by  any  algebraic  formula}  whatever. 

This  important  theorem  Avas  first  demonstrated  by  A])cl  in 
18::i5.  Previous  to  that  time,  mathematicians  frc(|uetitly  at- 
tempted to  solve  the  general  equation  of  the  fifth  degree,  bu^, 
of  course  never  succeeded. 

This  restriction  apjtlies  only  to  the  f/rnrrnl  equation,  in 
which  the  coefficients  p^^  p„,  p^,  etc.,  are  all  represented  by 
soiiarate  algebraic  symbols.  Sucii  special  values  nuiy  bo 
assigned  to  these  coefficients  that  equations  of  any  degree  shall 
be  soluble. 

343.  The  problem  of  finding  a  root  of  an  equation  of  (ho 
higher  degrees  is  generally  a  very  complex  one.  If,  however, 
the  equation  has  the  roots  —  1,  0,  or  -\-  1,  they  can  easily  bo 
discovered  by  the  following  rules : 

I.  If  the  nl^chraic  stun  of  the  coefficients  in  the  equa- 
tion vanishes,  then  +1  is  a  root. 

»7 


418 


GENERAL    THEORY   OF  EQUATIONS. 


II.  If  tlic  smiv  of  the  coefficients  of  the  even  poicers  of 
X  is  cqiKd  to  that  of  the  coefficients  of  the  odd  poivers, 
then  —  1  is  a  root. 

III.  //  the  absolute  term  21,1  is  wanting,  then  0  is  a 
root. 

These  rules  are  readily  proved  l)y  putting  x=  +\,  then  a?  =  — 1, 
then  J'  —  0  in  the  general  ('(luntiun  {(i)  and  noticing  what  it  then  reduces 
to.     The  demonstration  of  11  will  be  a  good  exercise  for  the  student. 

Number  of  Hoots  of  Goiiertil  Eqiiatlou. 

J?44.  In  the  cqiuilion  {a),  tlic  left -hand  number  is  an  en- 
tire t'nnction  of  x,  which  is  c<[ual  to  zero  when  tlie  equation  is 
satislied.  Instead  of  supposing  an  e(iuation,  let  us  su])i)osc  x 
to  he  a  variable  quantity,  which  may  have  any  value  whatever, 
and  let  us  study  the  function  of  x, 

.7"  -{-p^x^-^  +  p^x^-^  4- ^-2)n-ix  -\-J)n., 

wiiich  for  brevity  avc  may  call  Fx. 

AVhatever  value  we  assign  to  x,  there  will  be  a  correspond- 
ing value  of  Fx. 

Example.    Consider  the  expression 

Fx=z  a^  —  7.^2  +  36. 

Let  us  suppose  x  to  have  in  succession  the  values  —  4, 
—  3,-3,  — 1,  0,  1,  2,  etc.,  and  let  us  compute  the  corre- 
sponding values  of  Fx.    We  thus  find, 


X  = 


4,     -    3,     - 


-    1,  0, 

0,     +  28,     +  30, 


G, 


7, 


Fx  =  —  140,     —  o4, 

Fx  =  +  30,     +  10,    0,     —  12,     -  14,    0,     4-  30,     +  100. 

We  see  that  while  x  varies  from  —  4  to  +8,  the  value  of, 
Fx  iluctuates,  being  iirst  negative,  then  changing  to  positive, 
then  back  to  negative  again,  and  linally  becoming  positive  once 
more. 

We  also  sec  that  there  are  tliree  special  values  of  .r,  namely, 
—  2,  4-  3,  and  +  0,  which  satisfy  tiie  cquatioli  Fx  =  0,  and 
which  are  therefore  roots  of  this  equation. 


GENERAL    TIIEORT  OF  EQUATIONS. 


411) 


then  0  is  a 


345.  Representation  of  Fx  hy  a  Curve.  In  Book  VIII  it 
was  sliown  how  a  function  of  a  variable  of  the  first  degree  might 
be  represented  to  the  eye  by  a  straight  Une.  Tiie  relation 
between  a  variable  and  any  functiou  of  it  may  be  represented 
to  the  eye  in  the  same  way  by  a  curve,  as  shown  in  Geometry, 
Book  VII.  We  take  a  base  line,  mark  a  zero  point  ui)on  it, 
and  lay  off  any  numl)er  of  equidistant  values  of  x.  At  eaeii 
point  we  erect  a  per[)endicular  j)roportional  to  the  corres])onding 
value  of  Fx  at  that  point,  and  draw  a  curve  througli  the  ends. 


The  fluctuations  of  the  vertical  ordinatcs 
of  the  curve  now  show  to  the  eye  the  corre- 
sponding fluctuations  of  Fx. 

AVhen  Fx  is  negative,  the  curve  is  below 

the  ba^e  line.     When  Fx  is  positive,  the  curve 

is  above  the  base  line. 

The  roots  of  the  equation  Fx  =  Q  are  show^n  by  the  points 

at  which  the  curve  crosses  the  base  line.     In  the  present  case 

these  points  are  —  2,    +3,    +  G. 

In  order  to  distinguish  the  roots  from  the  variable  quantity 
a?,  we  may  call  them  «,  (3,  y,  6,  etc.,  or  .r^,  x„,  .r.,,  etc.,  or  «,, 
flg,  ^3,  etc.,  the  symbol  x  being  reserved  for  the  variable. 

The  distinction  between  x  and  the  roots  will  then  be  this: 

X  is  an  independent  variable,  which  may  have  any  value 
whatever. 

Fx  is  a  function  of  x  of  which  the  value  is  fixed  by  that  of  x. 

f(,  ft,  r,  etc.,  or  Xy,  X...  .r,,  etc.,  are  special  values  of  x  which, 
being  substituted  for  x,  satisfy  the  cfiuation 

Fx  =  0. 

Theorem.  An  equation  with  real  coefficients,  of  irhieh 
the  degree  is  an  odd  number,  must  have  at  least  one  real 
root. 


420 


GENERAL    TIIEORT  OF  EQUATIONS. 


U 


■liji 


Proof.  1.  When  w  is  odd,  X^  will  have  the  same  sign  (-|- 
or  — )  as  a;. 

2.  So  large  a  value,  positive  or  negative,  may  be  assigned  to 
X  that  the  term  x''^  shall  be  greater  in  absolute  magnitude  tliau 
all  the  other  terms  of  the  expression  Fx.  For,  let  us  put  the 
expression  Fx  in  the  form 

If  we  suppose  x  to  increase  indefinitely  cithe  in  the  posi- 
tive or  negative  direction,  the  terms  ~ ,  -^ ,  etc.,  will  all 
approach  0  as  their  limit  (§  303,  Th.  I).    Therefore  the  expression 

1  +  —  +  Sv  +  etc.  will  approach  unicv  as  its  limit,  and  w'" 

X       x^  ^ 

therefore  be  positive  for  large  values  of  x.  both  positive  and 
negative.  The  wliole  expression  will  then  have  the  same  sign 
as  the  factor  x"'.  and,  7i  being  odd,  will  have  the  same  sign  as  x. 

3.  Therefore,  between  the  value  of  x  for  which  Fx  is  negative 
and  that  for  which  it  is  positive  there  must  be  some  value  of  x 
for  which  Fx  =  0,  that  is,  some  root  of  the  equation  Fx  =  0. 

For  illustration,  take  the  preceding  cubic  equation. 

Cor.  Aji  equation  of  odd  degree  has  an  odd  nuniher 
of  real  roots. 

For,  as  Fx  changes  from  negative  to  positive  infinity,  it 
must  cross  zero  an  odd  number  of  times. 

340.  Theorem  I.    //  ice  diride  the  cxprcasion  Fx  hy 

X  —  a,  the  remainder  will  be  Fa,  or 

Remainder  =  n"  -{-  p^a'^~^  4-  ])»fi^~^  +  .  .  .  .  +  pn- 
Special  Illustration.     Let  the  student  divide 

3^  -|_  5x^  _|_  'Sx  +  1 

by  r  —  a,  according  to  the  method  of  §  00.  lie  will  find  the 
remainder  to  come  out 

«3  +  5^2  +  3rt  4-  1. 


' 


m 


same  sign  (  + 


■  be  assigned  to 
laguitudo  than 
let  us  i)ut  the 


ic    in  the  posi- 

etc,  will  all 

the  expression 

limit,  and  w*". 

1  positive  and 
the  same  sign 

:ame  sign  as  x. 
Fx  is  negative 

•me  value  of  x 

ation  Fx  =  0. 

ation. 

odd  number 

re  infinit}'^,  it 
cssioJi  Fx  hjj 

+  Pn- 


will  find  the 


GENERAL    TnEORT   OF  EQUATIONS. 


421 


Gencrcd  Proof.     When  Ave  divide  Fxhy  z—  a,  let  us  put 

Q,  the  quotient ; 
E,  the  remainder. 

Then,  because  the  dividend  is  equal  to  the  product,  Divi- 
sor X  Quotient  -f  Remainder, 

{x  —  a)Q  +  E  =  Fx. 

Two  tilings  are  here  supposed: 

1.  That  tliis  equation  is  an  identical  one,  true  for  all  values 
of  a:.  This  must  be  true,  because  we  liavc  made  no  supposition 
respecting  the  value  of  x. 

2.  '''hat  we  have  carried  the  division  so  far  that  the  remain- 
der li  does  not  contain  x. 

Because  it  is  true  for  all  values  of  .r,  it  will  remain  true 
when  wc  put  x  =  a  on  both  sides.     It  thus  reduces  to 

E  =  F{a), 
which  is  the  theorem  enunciated. 

The  value  of  x  being  still  unrestricted,  let  us  in  dividing 
take  for  a  a  root  a  of  the  general  equation  Fx  =  0.  Then, 
by  supposing  x  =  a,  the  equation  (a)  will  be  satisfied,  or 

Fa  =  0. 

Therefore  if  we  divide  the  general  expression  Fx  by  x  —  a, 
the  remainder  Fic  will  be  zero.    Hence. 

Theorem  II.  //  wc  denote  hy  «  a  mot  nf  the  equntiori 
Fx  =  0,  the  expression  Fx  will  he  exactly  d  I  visible  by 

X—  (I. 

Illustration.     One  root  of  the  equation 
.r3  — .r2  —  II.t;  +  15  =  0 
is  3.     If  we  divide  the  expression 

a^  —  .t"  —  11a:  -f  15 
by  re  —  3,  wc  shall  find  the  remainder  to  be  zero. 

347.  When  we  divide  /!r  by  x  —  «,  the  highest  pov/er  of 
X  in  the  quotient  Avill  be  x^~^.  Therefore  the  quotient  will  be 
an  entire  function  of  a;  of  the  degree  n  —  1. 


I 


422 


GENERAL    THEORY   OF   EQUATIONS. 


Illustration.    The  quotient  from  the  last  division  was 

X-  +  2  J'  —  5, 

•which  is  of  the  second  degree,  while  the  original  expression  was  of  tlio 
third  dejjree. 

If  we  call  this  quotient  Fj^:r,  we  shall  have,  by  multiplying 
divitior  and  quotient, 

Fx  =  {x  —  «)  FiX. 

Now  suppose  fi  a  root  of  the  equation 

F^x  =z  0 ; 

then  F^x  will,  by  the  preceding  theorem,  be  exactly  divisible 
by  X  —  (3. 

The  qiu)tient  from  this  division  will  be  an  entire  function 
of  X  of  the  degree  n  —  2.  This  function  may  again  be  divided 
by  X  —  y,  rcj)reseiitiiig  by  y  the  root  of  the  equation  obtained 
by  putting  the  function  ccjual  to  zero,  and  so  on. 

The  results  of  these  successive  divisions  may  therefore  be 
expressed  in  the  form 


Fx  =  (x  —  «)  F^x  ....  (Degree  w  —  1), 

Fyr  =  {x  —  (3)  I^\x (Degree  n  -  2), 

F^x  =  {x  —  y)  F^x  ....  (Degree  7i  —  3), 

etc.  etc.      etc. 


0) 


Since  the  degree  is  diminished  by  unity  with  every  division, 

we  shall  at  length  have  a  quotient  of  the  first  degree  in  x,  of 

the  form 

X  —  e, 
e  being  a  constant. 

Then,  by  sul)stituting  in  the  equations  (1)  for  each  func- 
tion of.*'  its  value  in  the  C(|uation  next  below,  we  shall  have 

Fx  =  {x-  «)  {x  -  (3)  {x  -  r) {x  -  «•), 

the  number  of  factors  being  equal  to  the  degree  of  the  original 
equation.     Hence, 

Theorem  I.  Firi*i/  entire  fun cf inn  of  x  of  the  nth 
decree  may  be  divided  into  n  factors,  each  of  the  first 
dc'Jrec  in  x. 


Ml 


vs. 

3 

ssion  was  of  the 
y  multiplying 


ictly  divisible 

lit  ire  function 
lin  be  divided 
ition  obtained 

r  therefore  be 


0, 


(1) 


cry  division, 
gree  in  x,  of 


o 


)r  each  func- 
hidl  have 

tlie  original 

of  the  nth 

if  the  first 


NUMBER    OF  ROOTS. 


423 


Since  a  product  of  several  factors  becomes  zero  whenever 
any  of  the  factors  is  zero,  it  follows  that  the  equation 

Fx  =  0 
will  be  satisfied  by  putting  x  efjual  to  any  one  of  the  quantities 
€t,  /3,  y,  .  .  .  .  e,  because  in  either  case  the  product 

(^_«)(,._^)(.,_y)..  ..(.,_.) 

will  vanish.    Therefore  the  quantities 

«,  (3,y, e, 

are  all  roots  of  the  origintd  e([uati()n  Fx  =  0.    Hence, 

Theorem  II.  ^l/i  algebraic  eqicatiuii  of  the  yi''*  dej^ree 
has  n  roots. 

"We  have  seen  (§  105)  that  a  ciuadratic  equation  has  two 
roots.  In  the  same  way,  a  cubic  equation  has  three  roots,  one 
of  the  fourth  degree  four  roots,  etc. 

Moreover,  a  product  cannot  vanish  unless  one  of  the  factors 
vanishes.     Hence  the  product 

Fx    or     {x  —  (c)  {x  —  (3)  {x  —  y) {x  —  e) 

cannot  vanish  unless  x  is  equal  to  some  one  of  the  quantities, 
€c,  p,  y,  .  .  .  .  e.     Hence, 

,ln  equation  of  the  n*^  decree  can  have  no  more  than 
n  roots. 

348.  We  may  form  an  equation  of  which  the  roots  shall 
be  any  given  quantities,  a,  b,  c,  etc.,  by  forming  the  product, 

{x  —  a)  {x  —  h)  {x  —  c),  etc. 
Example.    Form  an  equation  of  which  the  roots  shall  be 

-  1,     4-1,     1  +  2/,    1  -  2i. 
Solution.     W'c  form  the  product 

{x  +  1)  (x  _  1)  (.r  -  1  -  20  (.c  -  1  +  20, 

which  we  find  to  be 

X*  —  3.r3  +  4a:2  -f-  2x  —  5. 

Therefore  the  reciuired  equation  is 


J!| 


424 


GENERAL    TIIEORT  OF  EQUATIONS. 


•  EXERCISES. 

Form  equations  with  tlic  roots: 

1.  2  +  a/3,    2  —  \/3,     —  2,     +  1. 

2.  3  +  a/5,     3  —  a/5,     —  3. 

3.  2,     -  2,     4  +  a/7,    4  -  a/7. 

4.  1  +  a/3,     1  -  a/3,     1  +  a/5,     1  —  a/5. 

341).  When  we  can  find  one  root  of  an  equation,  then,  by 
dividing  the  e(iuation  by  x  minus  tluit  root,  we  «liall  have  an 
equation  of  lower  degree,  the  roots  of  which  will  be  the  remain- 
ing roots  of  the  given  equation. 

Example.     One  root  of  the  equation 

a:3  —  a^5  —  11a;  +  15  =0 
is  3.     Find  the  other  two  roots. 

Dividing  the  given  equation  by  a;  —  3,  the  quotient  is 

a-2  +  2a:  —  5. 

Equating  this  to  zero,  we  have  a  quadratic  equation  of 
which  the  roots  are 

—  1  +  a/G    and     —  1  —  a/C. 

Hence  the  three  roots  of  the  original  equation  are 

3,     —  1  +  a/C,     —  1  —  a/6. 

exercises. 

1.  One  root  of  the  efjuation 

a;3  _  3,^3  _  i4.y  ^  12  =  0 

is— 3.     Find  the  other  two  roots. 

2.  Find  the  five  roots  of  the  equation 

a:5  _  4^4  ^  l^^  ^  4^.2  _  13,^  _  q. 

(Compare  §  343.) 

350.  Equal  Roots.    Sometimes,  in  solving  an  equation, 
several  of  the  roots  rnay  be  identical. 
For  example,  the  equation 

.t3  _  (5a;2  +  12a;  —  8  =  0 


NS. 


COEFFICIENTS  AND   ROOTS. 


425 


ition,  tlion,  by 

shall  have  an 

je  the  remaiii- 


Lioticnt  is 

D  equation  of 


I  are 


an  equation, 


lias  no  root  except  2.  If  we  divide  it  by  a:  —  2,  and  solve  the 
resulting  (juadratic,  its  roots  will  also  be  2.  Hence,  when  we 
factor  it  the  result  is 

{x  —  2)  (x  —  2)  {x  —  2)  =  0. 

In  this  case  the  equation  is  said  to  have  three  equal  roots. 
Hence,  in  general. 

Hie  n  roots  of  an  cqnafion  of  the  n*^  dr^rre  arc  not  all 
ncccssctrll If  di/f event  from  each  other,  but  two  onnore  of 
them  may  he  equal. 

Relations  between  Coefficients  and  Roots. 

351.  Let  us  suppose  the  roots  of  the  general  equation  of 
the  m'^  degree 

a;"  +  PxX^-^  +  ;>oa;n-2  ^ _|_  ^  — ^  ^  ^_  ^^  _  q 

to  bo  «,  (i,y,....E. 

We  have  shown  (§  3-il)  that  these  roots  arc  functions  of 
the  coefficients  ^jj,  p.2,  ....  Pn-  To  find  these  functions  is  to 
solve  the  equation,  which  is  generally  a  very  difficult  i)r()blein. 

But  the  coefficients  can  also  be  expressed  as  functions  of 
the  roots,  and  this  is  a  very  sim})le  process  which  we  have 
already  performed  in  some  special  cases  by  forming  eciuations 
having  given  roots  (§  348). 

If  we  form  an  equation  with  the  two  roots,  a  and  P,  the 
result  will  be 

0  =  {x  —  «)  {x  —  /3)  =  a;2  —  («  +  i3)  a;  +  «i3. 

Comparing  this  with  the  general  form, 

a;2  +  PxX  4-^2  =  0, 


we  see  that 


Px  =  -  («  +  /3). 


Pz 


=  «3, 


a  result  already  reached  (§§  U)8,  100). 

Next  form  an  equation  with  the  three  roots,  a,  (3,  y. 
Multiplying  {x  —  «)  {x  —  (i)  by  x  —  y,  we  find  the  equa- 
tion to  be 

a^-{a+fi  -{-y)x^+  {(cl3  +Py  +  ya)  x  -  afiy  =  0. 


420 


GHNEltAL    TIIJJOIIY  OF  EQUATIONS. 


M 


So  in  tills  case,    ;?,  =  ~  (r<  -f  /?  +  y), 
/jg  =  «/3  +  /3y  -i-  y«, 

Adding  another  root  6,  we  find  the  result  to  bo 

;)i  =  -  («  +  /3  +  y  +  (5), 

;>3  =  «,3  +  «y  -f-  («^  +  /iy  +  (^'^  4-  r^J,  (2) 

jWg  =  —  «/3y  —  «^(5  —  «yJ  —  /jyd, 

Generalizing  this  process,  wo  reach  the  following  conclu- 
sions: 

The  coefRcient  ji-^  of  the  second  term  of  the  general  equa- 
tion is  c(iual  to  the  sum  of  the  roots  taken  negatively. 

The  coefficient  p^  of  the  third  term  is  ecjual  to  the  sum  of 
the  products  of  every  combination  of  two  roots. 

The  coefficient  jh  of  the  fourth  term  is  equal  to  the  sum 
of  the  products  of  every  combination  of  three  roots  taken 
negatively. 

Tiie  last  term  is  equal  to  the  continued  product  of  the  neg- 
atives of  the  roots. 

3/5'-^,  Sj/mniefn'c  FiDw/ionft.  It  Avill  be  remarked  that  the 
preceding  expressions  for  the  coeflicients  p^,  jk,  etc.,  arc  all 
si/N)  metric  fund  ions  of  the  roots  «,  (3,  y,  etc.     (§  250.) 

The  following  more  extended  theorem  is  true  : 

Theouem.  Fa'cvij  vatioiinl  syimnctvic  function  of  the 
roots  of  an  equation  may  he  expressed  as  a  rational 
function  of  the  coefficients. 

Example.     From  the  equations  (2)  we  find 

;7,2  -  2;?3  =  «2  4-  /3a  +  y2  +  d2, 

^PiVz  -  ih^  -  ^Jh  =  «'  -\-P'  +  y'  +  ^^' 

We  thus  reach  the  curious  conclusion  that  although  we- 
may  not  be  able  to  find  any  individual  root  of  an  equation,  yet 
there  is  no  difficulty  in  finding  the  continued  product  of  the 
roots,  their  sum,  the  sum  of  their  squares,  of  their  cubes,  etc. 

Tlip  p:t'n(?ral  demonstration  of  this  th(>orc'm,  and  the  niothod.s  by  which 
any  rational  synunetrical  functiou  of  the  roots  muy  be  determined,  are 
found  in  more  advanced  treatises. 


I 


I 


m 


vs. 


be 


A 


(2) 


owing  coiichi- 

gcncral  cquii- 

ively. 

to  the  sum  of 

al  to  tlie  sum 
e  roots  taken 

ict  of  the  neg- 

rkcd  tliat  <he 
etc.,  are  all 

.ctioji  of  the 
a  rational 


aUhoni::!!  we 
^({uation,  yet 
xluct  of  the 
cubes,  etc. 

liodf*  by  which 
ctermiut'd,  are 


DERIVED   FUNCTIONS. 


427 


Derived  Fiiiictious. 

353.  Dff.    If  in  the  expivssion 

we  substitute  x-\-7i  for  x,  and  then  cL'vclop  in  jHiwfrs 
of  //,  the  coefficient  of  the  first  power  of  h  is  called  the 
First  Derived  Function  of  x. 

To  find  the  First  Derived  Function.  Putting  z  -if  h  for  jr, 
the  result  is 

F{x  +  h)  =  {x-{-h)^-hPi{x  +  h)»-^  +  ....+pn-i{x+h)-^p^    (a) 

Developing  the  several  terms  of  tiie  second  member  by  the 
binomial  theorem,  we  have 

{x  4-  70"  =  .'i"  +  nz""-^  h  +  ^^  ^'\~  ^'  x^-^h'^  +  etc., 

(x  +  //)"-!  =  a-i-i  +  {u  —  1)  x^-^h  +  etc., 
(x  +  h)''-^  =  .T"-2  +  (;i  _  2)  x"-^h  +  etc., 
etc.  etc.  etc. 

Substituting  these  expressions  in  the  equation  {n)  and 
leaving  out  the  terms  in  li^,  h'\  etc.  (because  we  do  not  want 
them),  we  have 

F{x  +  h)  =  .r"  +  ;^,.T«-i  +  p.^xn-^  + +  p^-i  x  +  p„ 

+  [;2.r«-i  +  (;i-l)/)ia"-2  +  («-2);?2a"-H.  •  •  •  +/>»-i]  A 
4-  omitted  terms  imdtiplicd  by  h%  Ji%  etc.  {b) 

We  see  that  the  first  line  is  here  the  original  Fx,  while  the 
coefficient  of  h  in  the  second  line  is  by  definition  the  derived 
function.     So,  if  we  put 

F'x,  the  derived  function  of  Fx, 

we  have    F{x  -\-  h)  =  Fx  +  h  F'x  -\-  terms  x  h%  h\  etc. 

Let  the  student,  a.s  an  exorcise,  now  find  the  derived  function  of 

a^  +  8j3  -  5.1-2  +  7j.  _  9 
hy  the  process  just  followed,  commencing  with  equation  {n). 

Examining  the  coefficient  of  h  in  (J),  we  see  that  the  de- 
rived function  is  formed  by  the  following  rule  : 


428 


a  EN  Eli  AL    TUEOUY  OF  EQUATIONS. 


|: 


n 

t      I 


•» 


Miilti/)?!/  Cftch  term  hij  the  e,yj)nnent  nf  the  vdrifiWe  hi 
that  term,  and  dluilnisli  the  e.v/nmcnt  l)ij  icnifij. 

'''he  lust  or  coustiint  term  disappears  entirely  from  tlic  ex- 
pression. 

EXERCISES. 

Form  the  derived  function  of  the  following  expressions  : 

1.  a^  h  5r»  -{•  8x^  —  2^  —  x  +  1. 

Ans.  5x^  +  ^Ox-3  +  21^2  —  4^  —  1. 

2.  a;'  -  2.r»  —  S.t"  —  2x. 

3.  a*  -h  12.1-3  _  24^  +  if2  -f-  7. 

4.  X*  —  2r/a:3  ^  s^a.^'  +  a^x. 

5.  ofi  —  ^inz*  +  lOniic^  —  lo7nx\ 

Rkm,  Tho  student  should  obtain  tlio  result  by  aubstltutinpf  x  +  Ii  for 
h  in  each  equation  and  developing,  until  ho  is  master  of  the  process. 

854.  Second  Form  of  the  Derived  Funclion.  If,  as  bo- 
fore,  we  put  «,  i3,  y,  d,  etc.,  for  the  roots  of  the  equutiun 
Fx  =  0,  we  shall  have 

Fx  =  {x  —  fc)  (x  —  P)  (x  —  y)  .  .  .  .  {x  —  e).  ('•) 

Let  us  form  the  derived  function  from  this  expression. 
Putting  X  -]-  h  for  x,  it  will  become 

{h  +  x^  (c)  {7i  -\-  X  —  f3)  {h  -^  x  —  y) {h  ■}-  x  —  e). 

Studying  this  expression,  and  forming  the  products  which 
contain  h  when  three  or  four  factors  only  are  included,  we  sec 
that  the  cocflicient  of  the  h  in  the  first  factor  is  (x—fi)  {x—y) 
. .  . . ,  in  the  second  factor  {x — «)  {x — y). . . . ,  etc.  That  is, 
the  total  coefficient  of  h  will  bo 

{x  —  (3)  {x  —  y)  .  .  .  .  {x  —  e),  omitting  first  term  ; 
■{■  {x  —  a)  {x  —  y)  .  .  ,  .  {x  —  e),  omitting  second  term ; 

etc.        etc.  etc. 

+  {x  —  a)  {x  —  (3)  (x  —  y)  .  .  .  .    omitting  last  term. 

But  comparing  with  (r),  we  see  that  the   first  of  these 

.     ,     .       Fx       .,  ,    .       Fx 

products  IS  

*  X  —  a 

Fx 


,  the  second  is 


x  —  (3 


,  etc.,   to  the  last. 


which  is 


X  —  e 


llcnce. 


1 


N8. 


DEIUVEI)    FUNCTIONS. 


429 


le  t'n viable  in 
nit  11 . 

y  from  tlie  cx- 


ixprcssions : 
;2  _  4^.  _  1. 


rx  =  -^^-  + 


I     — 7i  H 4-  •  •  •  •  H •     ('«) 


tltutinR  x  +  h  for 
tho  process. 

m.     If,  Jis  1)0- 
t'  the  equation 

•  -  ^).  (") 

fpression. 

/i  4-  .T  —  e). 

roducts  which 
lutled,  we  sec 
(x-ii)  {x-y) 
etc.     That  is, 

first  term  ; 
second  term ; 

last  term, 
first  of  these 

to  the  last, 


JUnslralioit.      Let  us  take  once  more  the  expre.s.sion   of 

§344, 

Fx  =  3if^  —  7a-3  +  3G, 

of  which  the  tiiree  roots  are  —  2,  3,  and  C.    Its  derived  fuuc- 
tiou,  hy  method  (1),  is 

Expressing  Fx  as  a  product  of  factors,  it  is 
Fx=z  {x-\-  2)  {x  —  3)  (x  —  G). 

By  {d)  the  derived  function  is 

{x  -  3)  (.6-  _  G)  +  (a;  +  2)  {x  _  C)  +  (x  +  2)  (a:  -  3), 
■which  reduces  to  Zx^  —  14a?, 

the  same  value  as  by  the  iirbt  method. 

355.  Theorem  I.  IVhcii  the  derived  function  ir,  pofii- 
tive,  the  oi'igiual  function  increases  with  x ;  when  it  is 
negative,  the  function  decreases  as  x  increases. 

Proof.  When  we  increase  x  by  the  quantity  h,  Fx  is 
changed  to  i^(a;  -f  h),  and  is  increased  by  the  dilference 

F{x  +  h)  —  Fx. 

But,  by  {h)  and  {h'),  we  have 

F{x  +  h)  —  Fx  ■=.  h Fx  -f  h"^  x  other  terms 

=  h  {F'x  +  h  X  other  terms).       (r) 

Now  we  may  take  the  increment  h  so  small  that  h  x  other 
terms  shall  be  less  than  Fx,  and  then  F'x  4-  k  xother  terms 
will  have  the  same  sign  (+  or  — )  as  F'x. 

Then,  supposing  h  positive,  the  increment 

F{x  +  h)  —  Fx 

will  be  positive  when  F'x  is  positive,  and  negative  when  it  is 
negative. 

Theorem  II.  If  an  equation  has  cqiud  roots,  such  root 
ivill  also  he  a  root  of  the  derived  function. 


4C0 


GENERAL    THEORY  OF  EQUATIONS. 


Proof.  Let  /3  l)e  tlie  root  wliicli  /!»  =  0  lias  iu  duplicate. 
Then  when  Fx  is  factored,  it  will  be  of  the  form 

Fx  =  {x  —  «)  (.f  —  \V)  {x  —  (3)  {z  —  y)  .  .  .  .  {x  —  t). 

Kow  when  we  form  F'x  by  method  (2),  the  factor  {x  —  (3) 
will  be  left  in  all  liie  terms.  Theretore  x  —  (i  will  be  a  factor 
of  F'x.  Therefore,  whi'u  ./■  =  (3,  then  F'x  =  0,  so  that  (3  is 
a  root  of  the  Cfiuation  F  x  =  0. 

350.  If  the  equation  Fx  =  0  contains  no  equal  roots,  and 
if  we  sui)})ose  x  =  u  in  e(|uation  {d),  all  the  terms  except  the 
first  will  vani.sh,  because  the  common  numerators  /'.r  contain 
X  —  «  as  a  factor. 

In  the  case  of  the  first  term,  hoth  numerator  and  denomi- 
nator vanish  wheu  a:  =  «;  therefore  we  must  find  the  limit  of 

Fx 

when  x  approaches  «.     This  is  easy,  because 


X 


a 


X  —  a 


—  (•*"  —  ^)  (-^  —  y)  •  •  •  •  {^  —  «)• 


Therefore,  by  supposing  x  to  approach  «,  Ave  shall  have 

Fx 

Lim. (j-=a)  =  («  —  /3)  («  —  -}')....(«  —  £•). 

Therefore,  by  changing  x  into  «  in  ((/),  we  find 

F'lc  —  {a  —  d)  (rt  —  y) (rt  —  e). 

Ilencc 

Tlic  (Irvii'cd  ftuictinu  of  a.  rnnt  which'  has  iw  other 
root  ('(/iial  to  it  /.>■  tJie  rontiiiucd  prodiicb  of  its  dijfcrcuce 
from  all  the  other  roots. 

Significance  of  the  Derived  Function. 

3;">7.  Tiir.()Rr:>r.  Tltr  dcrirrd  function  cv/nrsscs  the 
r((ti'  of  increase  of  the  function  as  c,o})}j)arcd  willi  that 
of  the  varialAc. 

Proof.    The  equation  {o)  may  he  expressed  in  the  form 
F{,r  -f  //)  =  Fx  +  h  {F'x  +  Bh), 


ill; 


.  .1     I  mn. 


NS. 

LS  in  duplicate. 

.  (x  —  e). 

factor  [x  —  (i) 
,vill  1)0  II  factor 
0,  so  that  /i  is 

qual  roots,  and 
rnis  except  the 
jrs  Fx  contain 

)r  and  denonii- 
nd  the  limit  of 

iiso 

-0. 

shall  have 
(«  —  e). 
nd 

<ff<i  jin  other 
'Is  (lijj'crcuce 

ictioii. 

.vi>rrfisr.<i  the 
d  Willi  lltab 

the  form 


FORM   OF  MOOTS. 


431 


whore  7>P  is  the  sum  of  the  remaining  terms  of  the  develop- 
ment in  powers  of  h. 

\\q  then  have 

Increment  of  x  =  h. 

Corresponding  increment  of  Fx  =:  F{x  +  //)  —  Fx 

=  h{F'x+  Bit). 

Katio  01  these  nicrements,  — ^^^ — — , =  F  x  4-  JJ/t. 

h 

If  we  supjmse  the  increment  //  to  approach  zero  as  its 
limit,  the  product  Jilt  will  also  approach  zero,  and  the  ratio  will 
ap}>roach  F'x  as  its  limit. 

liut  this  ratio  of  the  increments  may  he  considered  as  the 
ratio  of  the  average  rate  of  increase  of  the  function  /'  to  that 
of  the  variable  x. 

Hence,  when  we  plot  the  values  of  Fx  by  a  curve,  as  in 
§  345,  the  derived  function  shows  the  slope  of  the  curve  at 
each  point. 

When  the  derived  function  is  positive,  the  curve  is  running 
np\\ard  in  the  positive  direction,  as  fron.  ;= — 3  to  .^•  —  0, 
and  from  x  =.  4-5  to  x  z=  -f-co. 

When  the  derived  function  is  negative,  the  curve  slo})e3 
downward,  as  from  :c  =  0  to  a;  =  +4. 

When  the  derived  function  is  zero,  the  curve  at  the  corre- 
sponding point  runs  parallel  to  the  base  line,  as  at  0  and  -f-4j. 
If  this  point  corresponds  to  a  root  of  the  e(|uation,  the  curve 
will  coincide  with  the  base  line  at  this  point,  and  will  there- 
fore be  tangent  to  it.     Hence,  from  §  350,  'V\\.  II, 

v'l  pair  of  cqudl  roota  of  an  cquafion  arc  ludicatctl  hij 
iJic  cicrue  toucJtuij^  the  base  line  wltJiout  intcrscctlmj  lb. 

Forms  of  the  Roots  of  Equation. 

;?i>S.  TnKOUEM  T.  Tnia^'i nary  roots  enter  an  equation 
wltli  rcat  cocffieicnls  in  fxdrs. 

That  is,  \i  a  +  hi  he  a  root  of  such  an  equation,  then 
a  — 1)1  will  also  be  a  root. 


432 


GENERAL    TUEOliY  OF  EQUATIONS. 


Proof.     Let 

a-n  ^  p^^.n-t  +  ^2.^«-2  + 4.  jr,^_^  r  +  ;)„  =  0       ( 1 ) 

he  tlie  oquation  wllli  real  cociVicients,  and  lot  us  suppose  tluiL 
a  +  bi  is  a  rout  of  this  equation.  It' we  substitute  a  4-  bi  i'ur 
a*,  we  shall  have 

»;»  =  ««  +  ?;«»-!  bi  —  ^^     7-  oP'-^  y^  —  (!!)  «""^  ^3/4-  etc. 

jOjiK""*  =  p^a^~^  +  p^a^'^bi  —  etc. 

If  we  substitute  all  the  terms  thus  formed  in  equation  (1), 

and  collect  the  real  and  imaginary  terms  separately,  we  shall 

have  a  result 

A  +  Bi  =  0 

(§  32-4),  A  signifying  the  sum  of  all  the  real  terms, 

n  (n  —  1) 


a**,       — 


2 


an-ib%      Pia'^-^     etc., 


»• 


and  Bi  the  sum  of  all  the  imaginary  ones. 

In  order  that  this  equation  may  be  satisfied,  we  must  have 

identically 

A  =  0,     B  =  0    (§  327). 

Next  let  us  substitute  a  —  bi  for  x.  Since  the  even  povers 
of  bi  are  all  real,  and  the  odd  powers  all  imaginary,  this 
change  of  sign  will  leave  all  the  real  terms  in  (1)  unchanged, 
but  will  change  the  signs  of  all  the  imaginary  terms.  Ilence 
the  result  of  the  substitution  will  be 

A  —  Bi. 

But  if  rt  +  hi  is  a  root,  then,  as  already  shown,  A  =  0 

and  B  =  0 ;  whence 

A  -  Bi  =  0 

also,  and  therefore  a  —  bi  is  also  a  root. 

Def.  A  pair  of  imaginary  roots  which  differ  only 
in  the  sign  of  the  coefficients  of  the  imaginary  unit  are 
called  a  pair  of  Conjugate  Imaginajry  Roots. 

Theorem  II.  In  the  expression  Fx  every  pair  of  eniiju- 
gate  i via ginary  factors  form  a  real  product  of  the  second 
decree  in  x. 


I 


vs. 

Pn  =  0       (1) 

1  suppose  thill 
lie  a  +  bi  for 


n  cfpiation  (1), 
ately,  we  shall 


ms, 
etc., 

we  must  have 


he  even  povers 
naginary,  this 
1)  unchani2;ed, 
terms.    LLeiice 


^howii,  A  =  0 


111  differ  only 
liijiry  unit  are 
>ots. 

\)airof  cnnju- 
of  the  second 


DECOMPOSITION  OF  RATIONAL   FRACTIONS. 


433 


Proof.     If  in  the  expression 

Ft  —  {.i:  —  a)  {.c  —  /3)  {x  —  y) {.c  —  e), 

we  suppose  u  and  /3  to  l)e  a  pair  of  conjugate  imaginary"  roots, 
which  we  may  represent  in  the  form 

«  =  rt  -|-  hi,        fi  ■=:  a  —  bi, 

then  the  product  of  tlie  terms  (x  —  a)  {.v  —  b)  or  of 

{x  —  a  —  bi)  (./•  —  a  -\-  bi), 

will  be  {x  —  rt)^  +  b\ 

or  x^  —  2az  +  d^  -f  b'^, 

a  real  expression  of  the  second  degree  in  ^. 

Cor.  Since  Fx  can  always  be  sci)arated  into  factors  of  the 
first  degree,  either  real  or  imaginary  (§  :3-i7,  Th.  1),  and  since 
all  the  imaginary  factors  enter  in  pairs  of  Avhich  the  pi'odiict 
is  real,  wc  conclude : 

Ercrii  piiUve  function  of  x  irifh  vcftl  roeffcirnis  inrrj/ 
he  divided  into  real  factors  of  the  first  or  second  decree. 

Decomposition  of  National  Fractions. 

359.  Dif.  A  Rational  Fraction  is  one  wliicli  may 
be  reduced  to  the  form 


.    +/ 


If  the  ex}H)nent  nt  of  the  numerator  is  equal  to  or  greater 
(ban  the  ex})onent  n  of  tlie  denominator,  we  may  divide  the 
numerator  by  the  denominator,  obtaining  a  (pioticnt,  ;ind  m 
remainder  of  wiiich  the  liighest  exponent  Mill  not  exceed 
n  —  ].     If  we  ]nit 

fx,  the  numerator  of  the  above  fraction  ; 
Fx,  its  denominator ; 
Q,  the  (luoticnt; 
0r,   the  reniaiiidcM' : 


fx 
we  shall  have,     Rational  fi-action  =  •,, 

28 


<?  +  >y      (§'-'<5-) 


434 


GENERAL    THEORY   OF  EQCATrONS. 


') 


(J  will  be  i\\\  entire  function  of  x,  witli  which  we  need  not 
now  further  concern  our.selves. 

The  i)rol)leni  now  is,  if  possible,  to  reduce  the  fraction 
^x 


Fx 


to  the  sum  of  ii  series  of  fractions  of  the  form 


A  B  C 

_j 1 [_ 

X  —  a       X  —  ii       X  —  y 


^  x-e' 


A,  By  C,  etc.,  being  constants  to  Ih;  determined,  and  «,  /?,  y, 
etc.,  being  the  roots  of  the  equation  Fx  ■=.  0.  Let  us  then 
suppose 

Fx 


_4_    _?_       r 

X  —  a       X  —  /J       X  —  y 


+ + 


E 


X 


(^) 


Multiplying  both  sides  by  Fx,  we  have 


AFx        BFx        CFx  EFx 

X  —  ((      X  —  0      X  —  y  X  —  E 


{^') 


We  require  that  this  equation  shall  Ik?  an  identical  one, 
true  for  all  values  of  x.  Let  us  then  supjiose  x  =  «.  TIkii 
because  by  hypothesis  «  is  a  root  of  ihe  expiation  Fx  =  0,  we 
have  /'««  =  0,  and  the  terms  in  the  second  member  will  all 
vanish  except  the  first.     If  there  is  only  one  root  «,  we  have 

(§  35r), 

T  •  ^'^  I" 

i^im. —  (jr=o)  z=  F  a. 

X  —  a 

Therefore,  changing  x  to  «,  we  have 

0fc  =  AF  fc, 

which  gives  A  =   ,,,   • 

^  F  (c 

In  the  same  way  we  may  find 

F^' 

F'y* 
etc.      etc. 


A 

V 

B  = 
C 


{<-) 


Substituting  these  values  of  A,  B,  etc.,  in  the  equation  {!>), 
it  becomes 


ys. 

1  we  need  not 

e  the  fraction 
a 
E 

\,  and  «,  /?,  y, 
Let  us  tlieii 


f 


B 


X  —  e 


EFx^ 
X  —  t 


{J>) 


{y) 


\  identical  one, 
'  a;  =  «.  Then 
ion  F.C  —  «>,  we 
ncmber  Avill  all 
root  a,  Ave  have 


(0 


he  c<^iuation  (/>), 


DECOMPOSITION   OF  RATIONAL    FRACTIONS.        435 


0a:  _      ^  <t>(t  0,'3  _^ 


:i^/>  -  -• 


Note,  'J'lie  critical  student  should  remark  tliat  in  the 
preceding  analysis  we  liave  not  proved  that  the  expression  of 
the  ratioiud  fraction  in  the  form  {h)  is  always  pussiblu,  hut 
have  oidy  proved  that  (/'it  be  possible,  Uicn  the  coillicients  J, 
J{,  (J  must  have  tlie  values  (').  To  prove  that  the  form  is 
possible,  the  second  member  of  {h)  may  be  reduced  to  a  com- 
mon denominator,  whicli  conimnn  denominator  will  Ijc  Fx, 
and  the  sum  of  the  numerators  e([uated  to  0.r.  liy  equating 
the  coellicients  of  tiie  separate  ])owers  of  .r,  we  sliall  have  n 
e([uations  to  determine  the  n  unknown  (pumtities  A,  B,  O, 
etc.  Since  ?i  (pumtities  can,  in  general,  be  made  to  satisfy  n 
e<puitions,  values  of  J,  //,  C,  etc.,  Avill  in  general  be  jiossible. 

It  will  be  instructive  to  solve  the  following  exercises,  lioth 
directly  and  by  the  common  denominator. 

EXAMPL.ES. 

I.  Decompose  -,_-^^, -^-3^.. 

We  have  alrea<ly  found  the  roots  of  the  denominator  to  bo 
—  2,  l},  and  'I.     Using  the  formuUe  {r),  we  find 

(l>x  =  -i.c^  —  :ir  4-  5, 

F.r  =  j^  —  r.',-2  H-  :]<;  =  {.r  +  'i)  (.<•  - ;])  {x  -  u), 

Fx  =  3./'^  -  14./- ; 

,c  =  -2,  fi  =  3,  y  =  0; 

(/)«  =  11),  0/3  =  14,  0y  =  59; 

F'i3  =  -  15,      F'y  -  24. 


i^'«  —  40, 
2.r2j- 3^  +_^ i:i izi _  "^ 


19 


14 


2.  Decompose 


%xi  _  7.r  +  3 


2a:2  _  7.,.  ^  3 


^  _  -^i-a  —  x-\r'l        {X 4- 1)  (a:—  1)  {x—l) 


Here  the  roots  of  the  denominator  are  —  1,  1,  an<l  2.     Let 
US  effect  the  decomposition  bv  tlie  following  method.     Assume 


u 


ill 


I  I  I 


'I 


n 


(I 


|i 


436  GENERAL    TUEORT  OF  EQUATIONS. 

(.r  -4-  1)  (.r  —  1)  (a-  —  y)        a-  +  i  "^  a;  —  1  "^  a:  —  "i'    ^'  ' 

Keduciiig  the  secuiid  member  to  a  commuu  denominator, 
it  becomes 

A  (.y2  -  3.r  +  2)  +  ^  (-t--  -  ^  --  -0  +  C{x^-\) 
(.r  +  1)  (x- -  1)  (a;  -  2) 

Since  botli  members  now  Inivc  tlie  same  denominator,  their 
numerators  must  also  be  e([ual.  E(iuating  them,  after  arrang- 
ing the  hist  one  according  to  powers  of  x,  we  liave 

{A  +  B^-  C)  x^  -  (3.1  +.5)  X  +  tA-'lB-C  ^  2x^  -  Hx  +  3. 

Since  this  must  be  true  for  all  values  of  x,  we  equate  the 
coefficients  of  .f  in  each  member,  giving 

A  +  /y  +  C  =2, 

3A  +  B  =  7, 

2A  —  2B  —  C  =  3. 

These  equations  being  solved  give 

A  =  2,        B  =  1,        C=  -1, 

Substituting  in  (d), 

2.ig  —  7x  +  3  _    ^      _1 1_ 

(x-  4_  i)  (.,•  _  1)  (.i-  _  2)  ~  ;r  +  1  "^  ar  —  1       .7  --  2* 

EXERCISES. 


Decompose : 
X  +  10 


I. 

3- 

5- 


a:2  — 4 

^^•5  _  U>a:2  —  S.r  +  lt> 


X*  —  ox^  +  4 


2a 


3? 


a' 


2. 

4- 
6. 


a^J  4-  8.r  +  4 


.T^  -I-  ir^  —  4a:  —  4 


a;-^  —  a2 


aW 


[x^  -  d^)  {x^  -  y^) 


JiOO.  When  the  e(|uation  Fx  =  0  has  two  or  more  equal 
roots,  the  ju'eeeding  form  fails,  because  all  the  terms  of  the 
second  member  of  {li)  will  then  vanish  when  we  sui)pose  x 
e(|ual  to  one  of  the  multi])]e  roots.  In  this  case  we  must  pro- 
ceed as  follows : 


I 


NS. 


c 


I  denominator, 

— —  ill    • 

ominator,  tlieir 
I,  after  arrang- 

ive 

we  e(|uatc  the 


X 


2 


Ix  —  4' 


C2—  Z»2) 


or  more  equal 
terms  of  the 

wu  su]»pose  X 
we  must  ])ro- 


DECOMPOtilTION   OF   RATIONAL    FRACTIONS.        437 


If 

wc  suppose 

0.r  .1 


Fx  =  (.>•  —  «)"*  {x  —  fi)"  {x  —  y)P, 
A,  A 


\m  "•"  (.,■  _  «)'«-!  "*"  (^  —  rc)'«-2 


/>  -  (.,•  _  «) 

(x  -  }  );>  "^  (:r  -  y)P-^  "^ "^  ;r  -  y 

ete.  etc.  etc. 


_9  "T  •  •  •  •  ~r 


J 


im    1 


+ 


In  the  ease  of  m,  n,  or  p  =  1,  this  form  will  be  the  same 
as  {b),  as  it  sliould. 

By  reducing  tiie  second  memlx^r  to  a  common  denominator, 
and  equating  the  sum  of  tiie  numerators  to  (px,  we  shall  have, 
as  Ix'fore.  a  number  of  equations  the  same  as  the  degree  of  x 
in  Fx. 

EXAMPLE. 


Decompose 


8a^  —  0.r3  —  2x  -r-  1 


of  which  the  roots  of  the  denominator  arc  —  1,  —  1,  1,  1,  3. 

Sn/i(fioii.     Because  of  the  roots  just  given,  the  expression 
to  which  the  fraction  is  to  be  equal  is 

A  A.  B  B.  a 


,      .iL,+ 


'    .  +  J^.  + 


(.r  —  1)2      x  —  \    '    (a;  +  1)^   '  a-  +  1    '   x  —  'Z 

Reducing  to  a  common  denominator,  and  equating  the  co- 
eilicients  of  the  powers  of  x  to  the  coefficients  of  the  corre- 
sponding powers  in  the   numerator   8./^  —  4;c2  —  2x  —  1,  we 

have 

J,  +/?,  +  C=       0, 

^  A,  +  A  -3Z?i  +  B  =       8, 

—  3^1  +  B^  —\B  —  HC  =  —  9, 

A^  —  3  J  +  77;,  +  5/y  =  —  2, 

2.1,  —  iA  +  2/>'i  4.  2Z^  +  6'  =  —  1. 

Solving  these  e<iuations,  we  fuid, 

A    =1,  B    =       2,  C  =  3. 

A,  =-2,  Il,=-h 


438 


(ll'NEUAL    TinCOUT  OF  EQUATIONS. 


I, 


Tlio  i;ivcii  Inictioii  is  tluMvfuiv  C'(»iiiil  to 
1  'Z  'i 


.-'-.-  + 


I  ;] 


(.,;  _  1)2      a;  —  1    '   (.r  +  1  )2      a;  -f  1    '   r  -  '^ 


EXERCISES. 


I.   Decompose  ^^/:^^' 


1  2 

A71S. T  + 


2. 


X—1 


X  —  I        {x  —  \  ) 
x^  —  2 


\i 


>•     a"^  _  .<a  +  a-  +  1 


'    9. 


4-    :;^-T  ,  • 


x^  +  ^.  ■    -  a;  -   1 


'I 


Greatest  Coniiiioii  Divisor  of  Two  Functions. 

lUil,  WluMi  we  luive  I  wo  e(|U;itions,  some  values  of  the 
unknown  (jUiintity  may  .sitisi'y  them  both.  They  are  then  said 
to  have  one  or  more  common  roots.  Such  CMinations,  when 
factored  as  in  s^  o47,  will  have  a  common  factor  or  divisor  for 
each  common  not.     Hence, 

TiiKoitKM.  T/ic  coinnioii  rants  of  two  equations  rnmj 
he  foiuid  from  their  greatest  coinvwn  divisor. 

Pu()1Uj:m.  To  find  the  greatest  eonnnon  dirisor  of  two 
('<lii<(ti()iis. 

This  problem  is  solved  by  dividing  the  two  polynomials  by 
the  methods  of  §§  90,  JiT,  aild  23:^. 

Example  i.    To  find  the  greatest  common  divisor  of  the 
two  polynomials, 

a:5  _  4^1  _(_  i2_^  _}_  J^.^a  _  i^^ 

and  x^  —  2.1-=^  -f  46-^  +  2.r  —  5. 

FIUST  DIVISION. 

ips  __  4,y4  _|_  10. ^;3  ^    4a;2  _  I'i^rj.  I  x^  —  2.T«  +  4.r^  +  2.r  —  5 


''->'■  \  X  —2 


2.?;'  +    H.?'3  +    ))xi  —    8x 


')  ,4 


4-    4.r—    8.1 


4r  +  10 


4:X^  -\-  lOr*  —    4a;  —  10  =  first  remainder. 


'iV-{?. 


+ 


{X  -  1)^ 


a-  +  1 


Functions. 

1  values  of  tho 
?y  tire  then  said 
juations,  when 
r  or  divisor  for 


indtions  may 
or. 

lii'isor  of  two 

)olynoniials  by 

divisor  of  the 


.r2  +  2.7-  —  5 


GREATEST   COMMON  DIVISOR. 

BECOND   DIVISION. 

a:4  _  2,^  _^    .1^.3  ^  o  J.  _  5  j  4.r3  -f-  lOa^^  — 4./  —  10 


439 


\x.- 


—  «J-3  +    5.1-2  .f  §.r  —  5 

^.c^  — *^  =  second  remainder; 


or. 


^^(.r^—  1)  =  second  remainder. 


In  the  next  division,  we  may  omit  the  fractional  factor  ^, 
because  every  value  of  x  wiiich  satislles  the  e<iuati(m  ;r^ — 1  =  0 
will  also  make  -Y  (•<-'' — 1)  =  0,  so  that  these  t'  niualions 
have  the  .^aifie  roots.  In  this  process  we  may  alvv.  ys  mltiply 
or  divide  the  terms  of  each  remainder  by  any  fac  >r  wh;ch  will 
make  their  coetlicients  entire. 


1 


TIIIUD   DIVISION. 

4.t3  +  10X2  _ 

Ax 

-10 

4./:3 

— 

ix 

lOarJ 

—  10 

10a:2 

—  10 

0 

0 

Ax  +  10 


emainder. 


Tlencc,  the  O.C.I).   of  the   two  functions  is  7?  —  \,  and 
their  common  roots  are   H-1  and  —1. 

This  result  may  also  be  reached  by  factoring;  the  given 
e([uations,  and  multii)lying  the  common  factors,  thus: 

a5  _  4.^4  ^  i2.,.3  ^  4^,2  _  i^x 

=zx{x  —  \)  {x  +  1)  {x  —  2  —  3t)  (.T  —  2  4-  3/), 

a:*  _  2a^5  +  4^:-  +  2.^;  —  5 

=  {x  —  1)  (.r  -f  1)  {x  —  1  —  2i)  (a:  —  1  +  20- 

We  see  that  the  common  factors  are 

[x-\){x-\-  1)  =  a;2-l. 


440 


UENKIUL    TI/KOIiV   OF   hQLATlOAi^. 


u 


'I'hi'  rules  for  tlirowiuy;  out  factors  from  divisor  or  ilividciid 
;uv  ;h  follows: 

I.  Jj'  hotli  'jUuiL  polijiKuniaU  cuntaiti  the  same  factor 
in.  (lit  their  terms,  remove  this  factor,  and  after  the 
(1.  ('.  I),  of  the  remainin'J  factors  of  the  two  polynomials 
is  found,  midtiplij  it  hi/  this  factor. 

Prnitf.  If  (I  Ikj  such  a  factor,  and  X  and  V  the  (juoticnts 
after  this  factor  is  removed  from  the  two  polynomials,  the  lat- 
ter, as  given,  will  be 

aX    and     aV. 

Since  a  is  now  a  common  divisor  of  both  given  polynomials, 
if  we  call  D  the  (J.C.I),  of  A' and  Y,  it  is  evident  that  aD  will 
1)0  the  (J. CD.  of  aX  and  aY. 

II.  Ally  factor  cominoii  to  all  the  terms  of  any  divi- 
sor, and  not  contained  in  the  dividend,  may  he  thrown 
out. 

Pro'if.  If  this  factor  were  any  part  of  the  G.C.  D.  sought, 
it  would,  by  §  'i'-Vl,  be  a  factor  of  each  dividend.  Since  the 
oidy  factors  we  recjuire  are  those  of  the  G.C.D,  factors  in  a 
divisor  only  may  be  rejected. 


li 


EXERCISES, 


'I 


I 


Find  the  G.C.D.  of  the  following  polynomials: 

1.  .r<  —  1  and  ./•  —  1. 

2.  .r^  —  1  and  .H  —  1. 


a^ 


"la^ 


a^  -f-  3«2  —  2n—ir)  and  a*—a^—^a^^a  +  5. 

4.  25.7^  +  5.v=*  —  X  —  i  and  20.^^  +  .r^  —  1. 

5.  a*  -\-  2(1^  +  9  and  r«»  +  ^ri^  —  Ort  —  9. 

6.  w3  +  3/yi2  +  ;im  +  1  and  in'-  —  L 

7.  .r»  —  ai-3  +  2l.c^  —  2Qx  +  4:  and  2x^  —  Ux^-}-21x—10. 

8.  a^  +  ft*  —  a  —  1  and  rt'  -f  a'''  —  a  —  1. 

iMVi,  The  given  polynomials  may  be  functions  of  two 
or  more  symbols,  as  in  §  9T.  AW-  then  arrange  them  accoril- 
ing  to  the  powers  of  one  of  the  symbols,  and  perform  the  divi- 
sions by  the  precepts  of  §  97. 


iUi  ii 


i  1 


m 

)v  or  dividcml 

same  factor 
d  after  the 
pol  Tjnoifilals 

the  quotients 
iiiials,  the  lut- 


1  polynomials, 
t  that  aD  will 


of  any  divi- 
ty  be  thrown 


.CD.  sought. 
Since  the 
factors  in  a 


'— 4a2_rt^_5. 


.1-2^  21a:— 10. 

ions  of  two 
(hem  aceonl- 
jrm  the  divi- 


anicATEr^r  common  divisoh. 

Ex.     Find  tlic  j^reutcst  common  divisor  of 

jcr^  —  (ix^  -\-  a  {b  +  c)  x  —  abc  —  b.i^  —  cj!^  +  ^ -k 
j^  —  a.i^  —  «(/»  +  c)  X  —  (d)c  -\-  bx^  +  cx^  +  bcx. 


411 


and 


The  (juotient  of  the  first  division  will  Iw  unity,  so  we  write 
the  two  functions  under  eacii  other,  thus: 

a*—       {a  +  b  +  r) x^  -\-  (nb  +  be  -\-  cn)x  —  abc 
^  ■\-  {—  a  -\-  b  +  c)  x^  —•  {ab  —  be  -f  ca)  x  —  abc 

—  'i  {b  -\-  c)  x'^  -\-       2  {ab  +  ac)  x  =  Ist  rem. 

Dividin;^  this  remainder  hy  —  'i  {b  ■}-  c),  we  have  the  next 
divisor.     Wc  then  perform  the  next  division  as  follows: 

x^  +  {—a  +  b-\-c)x^  —  {ab—hc-\-ca)x  —  abc  \  x^  —  ux 

^  -  'f^ I  X  -h  {b-k-c) 

{b  -\-  c)  .H  —  [ab — be  -f-  ra)  x  —  abc 
\h ^ c) .(^  —  [ab         -{-ca)  x 

bcx  —  abc  =  2d  rem. 

Dividing  this  by  the  factor  be,  Avhicli  is  contained  in  all  its 
terms,  we  have  x — a  for  the  next  divisor,  which  we  find  to 
divide  the  last  divisor,  and  therefore  to  be  the  G.C.D.  required. 

EXERCISES. 

Find  the  G.C.D.  of 

T'  +  3brx  +  l^  —  c^  and  .r**  ■i-{e—b)x^-\-  {f?  ^bc^c^x 

a-3  +  ,3^/.r  +  a^  —  1  and  x^  —  [a-  —  2«)  x  -{-  n  —  I. 

{a-\-b-\-c)  (ab-\-bc  +  ca)  —  abc  and  a!^  -f-  ab  —  ac  —  be. 

.r»  4-  4«*  and  x^  —  2a^x  -f-  -kfi. 

x^  —  ax^  —  b^x  +  ab^  inid  x^  —  a\ 

g^  j^ffi  j^j^  —  ^abx  and  x^  +  Hax  +  a^  —  i^. 

a4  _  2x'i  +  2  -  ^,  +  ^.  and  .r»  -  2x^  4-  r  -  1 . 
x^      .-r'  x*      7^ 

.<r*  —  j^u  -f  ./•//•'  —  if  and  .r*  +  .'•''//-  +  //'. 


410 


f.'/:XKii.\L  Tn/:onr  of  equations. 


Traiisloriiialioii  of  K(iiiali<>iiH. 


:  I 


'I 


li  1 1 

1:    ■'      '       ■ 

ii    I: 

IMMi.   Dff.     An  ('(Illation  is  said  to  be  Transformed 

wlini  ji  sct'oiid  ('(imilioii  is  IoiiikI  \vlios(.'  roots  l)(.'ai'  a 
known  ivkition  to  those  of  tin?  givon  ('(^luation. 

1ii;m.  SoniL'liines  wo  may  kt  jihlc  to  llnd  a  root  of  the 
(raii,-«ronmHl  uijuulioM,  and  tiiciK'c  tlic  corrt'sjtomliiii,'  i-.M)t  ol' 
the  original  ('(|uuti()n,  more  ca.sily  iiiuii  by  a  direct  solution, 

PfjonLFM  I.  To  chdii^c  the  signs  of  nU  the  roots  of  an 
cqtKtfiun. 

Sohitwn.  By  changing  x  \\\U>  —x  in  a  given  c(iuation, 
the  signs  of  the  terms  containing  odd  powers  of  x  will  be 
cliangetl,  while  those  of  the  even  lowers  will  be  imehangod. 
Ilenee,  if  u  be  any  root  of  the  original  e(j nation,  —  «  will  be 
a  root  of  tlu'  e((Uation  after  the  signs  of  the  ulLernute  terms  are 
changed,     llenee  the  rule: 

Change  the  signs  nf  thn  alternate  terms,  of  odd  and 
even  degree,  in  the  equation. 

Prohli:m  ir.  To  diminish  all  the  roots  of  an  equa- 
tion hij  the  same  quantitij  It. 

Solution.     If  the  given  C(iuatioii  is 

X'l  +  ;;,  <;«-!  +  p^,,n-^  +  .  ...  +  ;,„-  (), 

and  if//  is  the  unknown  (|iiantity  of  the  re(|uired  e(iuation,  we 
must  have 

}j  =  X  —  h. 

Therefore,  x  =  ?/  -f  //. 

Sul)stituting  this  value  of.?;  in  the  equation,  it  will  become 


>/"  +  {Pi+>d>)!/"-'-\- 


/'2-\-0'-^)pJ'  +  Qf''  !//"-^-hc'te.    {a) 


When  Ji,  «,  and  the  ;/s  arc  all  given  (|uantities,  the  c(jefii- 
cieuts  of  y  become  known  quantities. 


I 


» 


l(L\8. 


OUH. 


!  Transformed 

B  routs  bear  a 
Lition. 

1(1  ji  root  of  (ho 
i^jtomlini;  I'.'ot  ol' 
lirccl.  solution. 

I  tlu  ruuts  of  an 

given  cMiiiation, 
,ors  of  X  will  be 

II  1)0  uiic'lianjjjod. 
ion,  —  «  will  1)0 
Loniuto  tirnis  uro 

'}is,  of  odd  (I lid 


ols  of  an  cqua- 


=  0. 


irod  oquiition,  wo 


n,  it  will  become 
titles,  tbo  coofli- 


i 
I 


N 


(IKS Kit M.    TIII'.OliY    or   EQUATIoys. 


EXERCISES. 


443 


1.  'Pninsfunn  (ho  o<|iialion  /^  —  \\x  —  4  =  0  into  one  in 
wlii(.'li  liu!  roots  jsliiill  be  loss  by  1. 

2.  'rranslurm  .r' —  .*i.c-  4-  b'ix  —  7  =  0  into  one  in  wliiob 
the  roots  shall  be  greater  by  0. 

'MW,  Ikcmoriuii  Tirins  from  Ef/Ufttionx.  'IMio  <jnantity  // 
may  be  so  (.hosrii  that  any  roi|iiiro(l  term  after  the  lirst  in  the 
traiisfornied  o(|ii!itioii  shall  vanish.  For,  if  wo  wi.sh  the  second 
term  of  the  eciuution  {u)  to  vanish,  we  have  to  supjiose 

p^  +  nh  =  0, 


wliieh  gives 


Wc  then  substitnto  this  value  of  h  in  the  o<|nation  (r/), 
which  gives  an  o(iualif»n  in  which  the  second  term  is  wanting. 

If  we  wish  the  third  form  to  vanish,  we  must  determine  h 
by  the  condition 

which  ro(|uires  the  sohition  of  a  quadnitic  er|uation.  Each 
eonsccutivo  term  is  one  degree  higher  in  the  unknown  (|uan- 
tity  h,  and  the  last  term  is  of  the  same  degree  as  the  original 
eijiiafion. 

This  nu'thod  is  i)rincipally  ajjplicd  to  nuiko  the  second 
term  disa])pear,  which  re(juiros  that  we  put 

KxAMi'Li:.  Make  t lie  second  (orm  disa]»|tear  from  the  fol- 
lowing I'liuation, 

.,.2  4-  p.c  +  7  =  0. 

Sohitio)!.     Hence,  n  ='i  and  //,  = />,  so  that 


//  = 


X  -  ij 


I' 


P 

■ 


'» 


4^W  (IKNIJUAL    Til  Hour    oF    IK^UATIOMS. 

Makiiiu  this  s'ilAstitution,  tlic  t'(jiiatioii  becomes 

//'^  -  f  +  7  =  0, 

wliieh  IS  thf  idiuiivd  e(|Uutioii. 

liE.M.  'J'liis  ]>n»r'ess  airurcls  tui  uclditiouul  elegant  ihethod  of 
solving''  the  (jiuuh'atie  eciuatioii. 
The  huit  equation  givc-s 


=  v/? 


I 


Thu  vtilue  of  X,  beiniT  e(iiial  to  //  -f  fi,  then  becomes 
whieli  is  the  correct  sohitinii. 

EXERCISES. 

Iteinovf  the  secontl  »erm  (Voni  tlie  lollowing  e((uations  : 

1.  .r'  —  (;.f-  +  «;./•  —  I  r=  0. 

2.  .H  —  -l./-''  H-  '.\/^  —  t^  —  0. 

J.     .'•=  —  .5./-'  4-  '^i^  +  '"i.^-  —  ;i.''  =  0. 
4.     x^  —  12.^5  -I-  ;ix3  —  J-  =  0. 

IiKM.  The  theory  of  the  above  process  will  be  readily  com- 
l>rehended  by  recalling  that  the  cocnicients  oi  the  second  term 
is  e(iual  to  the  sum  of  the  roots  taken  negatively,  or  if  «,  /3,  y, 
ttc,  be  tlio  roots, 

«  +  /^  -f  )  4-  .  .  •  .  -]-*■■—  — />,. 
ft  is  evident  that  if  sw  siilttract  the  arithmetical  mean  of 

all  the  roots,  that  is,   —     '  ,  from  each  of  them,  their  sum  will 
vanisii,  l)ecause 

u  f   ^^  4  >i   h    ^^ 


f-  )   4- 


n 


f  .-Ic 


1  


~V'«  +  '^  ..    =  ^ 


?i 


llenct'.  uheii  \\c  pul   //  —  -  '  for  x  \  \  the  equation,  tile  sum 
of  thf  roof-,  and  (Ikk  loiv  the   second  tciin.  vanish. 


lies 


rant  ihc'thod  of 


becomes 


r  e(iiiutiuns 


bo  readily  coni- 
hv  SL'Coiul  loriu 
V,  or  if  «,  /3,  y, 


ictical  iiu'an  «»!' 
,  their  mm  will 


71 


nation,  the  sum 
1  h. 


a  EN  ERA  L    TUF/HiY   OF   KQUATIOXS. 


445 


3G5.  PiiOBLEM.     To  fr/tns/'orm  (tit  ('(/Ufifion  ,s()  (lull  tlie 
roots  slmll  hi'  iiiiiUift/'cd  hij  a  gireii  factor  m. 

tSdliifimi.     Siiicc  the  roots  art-  to  be  miiltijtlicd  by /^/,  llie 

new  unknown  ([uantity   must   be  equal  to  mx.     So  if  we  call 

this  quantity  y,  we  have 

y  =  mx, 


which  "fives 


'  —  1L 

~  m 


Sul)stituting  this  in  the  general  equation,  it  becomes 

Multi[>lying  all  the  terms  by  m",  llu-  c(|nation  becomes 

.'/"  +  '";'iy"~*  +  >fi'^p.>!J^'~'^  -f  ....  4-  m"pn  =  0. 

Ileneo  the  rule, 

MuUiphi  the  coefficient  of  the  second  term  hij  m,  thot 
of  the  third  Inj  in\  (tud  so  on  lo  the  lost  term,  ir/iieh  irill 
J)e  mulfi plied  Inj  ,ii". 

If  the  roots  are  to  be  divided,  we  divide  the  terms  in  the 
same  order. 

EXERCISES. 

1.  Make  the  roots  of  x^  —  '^.r  -f  ;3  —  0  four  times  as  great. 

2.  Divide  the  same  nH)ts  by  2. 

8G(>.  riJoin.i'.M.  To  trotisforni  (in  eqiifdion  so  Ihut  its 
roots  siudl  he  sipiiired. 

t)olulion.     Let  the  given  e(|uation  be 

If  //  be  the  unknown  (juantity  of  the  new  equation,  we 
must  have 

V  —  A 

whicli  gives  ^  =  ±  //-• 

If  wo  substitute  :f  =: /y-'  in  the  given  e(|Uation,  it  may  be 
reduced  to  the  form 

//•      f     /'■.■//      I      /'4      I      (/'l//      I     P.\)  U'    ■-"    <•• 


440 


GJ'JNh'liA/.    TUKOliY   OF   KQUATIONS. 


}{'  we  sul)stitute  .i"  =r  —  y--,  the  result  will  be 

f  -f-  /'a//  +  JU  —  {Pi'l  +  Ps)  !/'  =  "• 
Since  the  viiliic  c»l'  t/  imist  .sitisry  one  or  the  other  of  (Iiesc 
0(|ii;iti()ns,  it  nin.-t  reduce  their  pniduet  to  zero;  we  therel'ni'c 
imihiply  tluMM   to<^efher.     Con.^iderin^'  them  as  the  simi  iiiid 
dillereiice  of  a  pair  of  e.\j)ressi()n.s,  the  product  will  he 

(//'  +  P^y  +  JUf  -  iPi!/  4-  PsYl/  =  0, 
or 

y'+{'^P2-Px')f+iP'z'+''^PA-''iplP3)!/'  +  (^P,Pi-Pi')!/+P4' 

=  0. 

EXERCISES. 

1.  Transform  the  ((uadratic, 

xi  —  5.r  +  0. 

of  which  the  roots  are  2  and  .'{,  into  an  o(|uation  in  which  the 
roots  shall  he  the  stjuares  of  2  and  3.  using  the  ahove  pnn-ess. 

2.  Transform  in  the  sanu^  way 

;i-^  +  I'^jf^  +  U.c  +  48  =  0. 

3.  Transforn 

./;5  _  4^.4  _  i();j:3  _|_  4o_j.2  _f_  O.J-  _  ;}0  z=  0, 


U 


Gonornlizatioii  of*  the  Pre<*e<liiij»:  Problems, 

J5G7.  TuoiiLKM.  (^fiven,  an  C(/it(ttioii  oj'  (un/  drgrce 
in  tin.  u iikinncii  (/Hanfifi/  :r  ; 

Ke(|ii;re(l,  to  /rtiiisJhrDi  f/u's  cqiuiHon,  into  (inotlnr  of 
It'll i ell  the  root  sluill  fjc  ii  ijircii'  Junction  of  u. 

Solution.  Let  y  he  a  root  of  the  re(|uired  C(iuation,  and  j'.r 
the  r(iv(>n  function.     Wc  must  then  have 

./■'■  =  //• 

Solve  this  c(|uation  s(t  as  lo  ohiajn  ./as  a  function  »»f  //. 
Siihstitiiti'  this  value  of  .r  in  Ihc  nHjiinal  e<|uation.  and  Inrni  as 
niai\v  f(|iialions  as  ihri'c  are  vahies  of//. 

The  pi'oduct  of  the^e  ciiUHlions  will  be  the  required  eiju;i- 
li'iii  in  //. 


I 
I 


I 


'ONS. 

he 

=  0. 

ic  oilier  of  llu'SC 
u  ;  wo  tlitTcroiv 
lis  the  sum  iiiid 
will  lie 

=  0, 

=  0. 


oil  ill  which  tlie 
.'  ahovc  process. 


=  0. 


I'robleiiis. 

oj'  (III  If  (Icjjrre 

\,ii)  (inothrr  of 
<'  J. 

'([luitiou,  and  // 


I  iuiu'tioii  of  I/. 
ion,  and  t'orni  as 

r('(juired  ('<ju;i- 


OENE^lM    TIIEOKY   OF   EQUATIONS.  447 

EXERCISES. 

1.  Traiit;fonn 

,c^  —  :./•  -f-  10  =  0 

feo  Ihat  the  roots  of  the  luw  ('(jiuitioii  .shall  be  'dx\ 

2.  'i'nuisforin        t^  ~  :U-'  +  ix  =  0 
so  tiiat  lilt'  n^ots  ^liall  be  ax  -f  b. 

,5.  '1'ijiii.sfonu  .'•^' —  !>r  +  IS  r=  0 

so  thai  the  roots  shall  be     .r-  —  3. 

o 

liosohitioii  of  Xunicricjil  Kquatioiis. 

J5(>«S.  ('(nircitient  invthoil  nf  com/n(/ii>(/  llic  nunivrivdl  rahic 
of  (III  (III  ire  function  (fxftr  ait  (issutiicii  value  (f  x. 

If  we  liave  the  entire  function  of  ,/■, 

F.r  =  ax^  -f  Ox^  +  cx^  -f  dc  +  C, 

\\c  may  jmi  it  in  the  ronu 

FX  —    \\{((.r  +  f>)x  +  (■]  X  4-  (/\  X  -f  c. 

Therefore,  if  we  put 

ax  f  /.'  ■=.  //,  f)'x  -f  ''  =  ('', 

c'x  -r  >l  =  'f,  (I'x  -\-  c  =  c, 

we  shall  have  Fx  —  c'. 

NiiiucrictfJ  Fxawplc.     Compute  the  vahu's  of 

Fx  —  •*./-"'  —  ;].'  *  —  (ir*  -f-  S.i-  —  1) 

for  X  =  '.)  and  .'•  =::  —  'i. 


We  arrange  the  work  thus; 


Coetlicient: 


'Z       -; 


t  f   H         -     \) 


Prnd.  by  (./■  =  :$),  +<>         -I-'.'         +0         +^7         -fKT) 


II 


encc 


'or  X 


lleucc 


-f-:{         ^:{  +0  +:{.'>        + 

Z':}  —  !m;. 

— :{      -   (I  0  +  s 

_4       4-14  —  k;  ^'.Vi 

— ;      -+-  8  —10  -I--W 


•  SO 

SI) 


448 


Oh'NK/UL    TULOJiY   oF   h:f^VATIt\ei. 


Tliis,  if  will  be  noticed,  is  a  more  ronrenir'nt  prcx^ss  t'lan  thui  of 
fiiriuiii^-  till'  -lovvurs  «>>  x  ami  iiiultijilyitig  and  a<Iduig. 

iiOlK  Ifdi'iwj  an  f'H/irc/u/tcfion  of  x^and  jmlliufj  x~i'-\-h, 
it  is  required  lu  dcrel'/j)  Ihc  function  in  jt'twer^  of  It, 

It  will  be  rcinaikcil  iluit  tliis  problem  is  8iih>iaiitia]ly  identical  with 
iha*  of  ^  .U>2.  niid  the  fiulution  of  this  will  l*e  the  iudution  of  tlie  former. 
lint  in  the  I'ornier  ca.se  //  was  su|>i>o**ed  to  l*e  a  givn  (juantity,  whenas  it 
ifi  now  the  unknown  tjuautity  C(>rre»[)oniliQ^  to  ^  iu  the  former  problem. 

Example  OF  Til K  I'iioblkm.     If  wc  Uuvc  the  expression 

and  put  x  =  2  i-  h,  it  will  become,  by  developing  the  sepa- 
rate terms, 

F<^i-\-h)  =  U^  +  15/<*  4-  3G//  +  32. 

GEXERAb  HiLi:  FOR  THK  pROfE^s,  First  rom/mtr  the 
value  of  Fr  Inj  the  process  rnt/tfnffet!  in  $  .'JOO. 

Then  repeat  the  process,  usi/tfi  the  HUfresaire  snw<  oh- 
tdiiieii  in  the  /ir.-it  process  inste/nJ  of' thr  mrrr spout! !}!<i 
coejjieieuts,  and  st(tppiii<J  one  trnn  In  for i  th<  last.  The 
result  trill  he  the  eocjjirimt  of  h. 

liepeat  the  process  with  the  new  taims.  stojifiinij  j/rt 
one  term  sooner.     The  result  will  he  the  lUH-Jficicnt  oj  h"^. 

Continue  the  repetition  until  irr  hare  the  first  term, 
onlif  to  operate  upon,  uhick  will  iltfclf  be  the  coefficient 
of  the  hii>liest  })ou:er  of  h. 


'» 


Ex. 


Til 


u'  example  ;','>« »vi'  ;,'i\r?i  ;.^  performed  a.--  follow!- 


('oetlicients, 
PnKhict  l)y  )\ 

FirHt  Hum' , 
Sen»ul  j>i.idut'tH, 

Second  sums, 
Third  proiluet, 


f3 


11 


15 


14 
14 


f4 


38 


Result, 


/'('iH-A)  -'-  2^5+  Vth   •  a6/r  -  32. 


Ex.  2.      Ill  llie  riiiictii'ii, 


let  I'.-  pHl 


Fx  =z  t(^  -  :.r«  4-  r>:r»  -  ix*  +  e,x  —  8, 


// 


7.    111(1  cvpres-'  (III-  n 


■»iiil  i 


U  jKfUiT*  of//. 


ucesB  t'mn  thai,  of 

Silly  identical  witlj 

i*n  of  tin-  foiiucr. 

lantity,  whereas  it 

iormer  proUIeiu. 

lie  cx})rL'isi!jiim 
I'ing  the  scpa- 

coin  pate  the 
». 
^nivc  s^ntK  iih- 

fw  last.     The 

yfoppin;^^  ijct, 
firicnf  oj  li'^. 
he  p'j'st  term 
he  coefficient 

.•<1  as  follows: 
•  4 


s. 


GENiniAL    TllEOllY    OF  EQUATION.\ 


440 


■  vvi.'rs  oi'//. 


('ofincients,             2 

ft 
—  i 

+n 

46 

-P 

i'roiliicts  Ity  3, 

0 

-;j 

+  6 

+  12 

+  .>4 

First  sums. 

-1 

+'i 

+  4 

+  1« 

+  Kj 

SctoikI  productH, 

+  6 

+  15 

+  51 

+  u;5 

Sfcoiid  Hiiiii.s, 

Ts 

+  17 

+  55 

~1»3 

Tiiird  products, 

6 

y;i 

150 

Third  riuitiu, 

11 
0 

17 
6 

£'3 

no 

51 
101 

205 

Rosult,         F(3  +  /0  =  3/t'  +  23A^+I01A^  +  205A'+183//  +  40. 

EXERCISES. 

1.  Compute  U^  -\--ZU*  +  l(il//3+  •^or)//2  +  18;3/<-f-4r),  when 

2.  Compute  /'—  *..f  -f  ^  for  .c  =  —  }  -f-  //,  _  3  -j.  //,  etc., 
to   +  :j  -4-  //. 

Proof  of  the  Preceding  Procrss.    if  we  develop  tlie  ex- 
pression 

iiiul  rollect  the  roi'iricients  of  like  powers  of  //,  we  shall  lind 

Coef.  of  //"      —  n, 

//«-'  =  nar  +  b, 

lin-i^i^'^nr^^^-i^n-Whr^c,  (J, 


[•i)'"''  ^("  l   ')^'^-'  +  ("  -'i)rr-^il, 


A" 


/<»-"  =  y\ar'  +  r  _  Ahr'-^  -f  (^J  __  J<^/^^  +-  vw. 

Now  ex{iniinin<;  Kx.  2  precediufj,  it  will  be  seen  that  wc  can 
make  I  lie  ('oniputati«;n  by  eolunins,  lirst  eomputin;^  the  wliole 
left-band  (•i>luiiiii  and  tliiis  olMainiiij]f  the  eoetlieient  of  A'*  '. 
ilien  ('()ni|)ntin;/  the  next  cobinm,  tbus  obtainiiii;  the  eoclli- 
eient  of  /<"  '^,  and  so  on.  ('oinnu'n('in<r  in  this  way.  and  usin^ 
the  literal  eoetrieiejits,  a,  i,  r,  ele.,  and  the  literal  iaetor  /•,  we 
«haii  iiavc  the  rej?ult.s: 


450 


UENEUAL    rilhlOIiy    OF   EQUATIONS. 


Ii< 


h 

ar 

6 

br 

(ir  -t-  h 
ar 

ar^  + 
2ar^  + 

br  +  c 
br 

2(ir  +  b 
ar 

Sar^  + 

Ur  +  0 
br 

liar  +  b 


Car''  +  '.ibr  +  c 


liar  -(-  b 


(!!)"'"'  -\-  {>i  —  \)  br  i-  c. 


If  M  is  the  (1('<j:iv('  of  the  (.'(|U;itioii,  tlioii,  by  tlio  ])roc(.'ding 
pnttrss,  wo  shall  ailtl  tlic  product  i(r  to  b  a  tinu'S,  tlic  n  sepa- 
rate .sums  \m\\\f 

((r\-b,     "Zar  +  b,     ',]ar-\-b,  ....  nar-{-b. 

To  form  the  second  cdhimn,  wo  inultii)ly  each  of  those 
sums  oxcopt  llio  hist  hy /*,  and  a<Ul  thom  to  the  coollioiouL  <•. 
Tho  tonus  in  ar  ad(h'd  i)oin<,'  ar-y  'iar^,  {\ar-,  oto. ,  the  sum 
will  1)0  ( 1  -f- ,'  f ;{ + +  ''  —  !)  <ir^.     Tlie  coollioiont  is  a  figu- 

rato  i>uml)or  o(|ual  to  -A^-i-^  (§^  'Zi^i],  '^ST).     Tho  sum  of 

tlu'  oDi'llicionts  of  Ir  is  ;/  —  1,  hcoause    thoro   aro  ;/  —  1   of 
Ihi'UJ  usod.  caih  o<pi:d  to  unity.     Thorrfore  tho  linal  result  is 


(!;)'"^+  i»-  \)br  +  c. 


wiii.'h  wi'  have  found  to  he  the  eot'llieient  of  A"    '. 

Fn  this  second  column  the  partial  sums  or  cooillcienis  of 
(tr'^  are 

1,    1+2  — y,    1 -f J +  ;}  =  <•»,  otc,  to    l-(-2  +  n  +  .. ..+(;/  — 2). 

Tlvjrofor:  "he  numbers  succosively  adrled  to  form  tho  co- 
ollieirtUi  .  L  (//■'  in  tho  third  column  aro  1,  1+3  r=  4,  1+3  + (J 
—  10,  etc.  The  '.'oetlieieiits  of  Itr^  will  Ix;  the  s;imo  as  these  of 
ar^ '\\\  \\\v    itUtiiU;  next  pncodiuir. 

CoutiniiiiijT  the  ju-oec^s,  v/e  see  that  the  cooflicionis  aro 
formed  l)y  tsU'^cossive  addition,  as  in  tho  following  table,  where 
each  nnnd>or  is  the  sum  of  the  one  al)ovo  it  plus  the  one  on  its 


I 


'IONS. 


+  c 


+  c 


4-  c 


n  —  1)  br  +  c. 

nic's,  tlie  /<  sepa- 

/;•  +  h. 

ly  t'iieh  of  these 
lilt'  coeHicient  c. 
•',  etc.,  tlio  sum 
llic'ient  itj  a  ll^^u- 

).     The  sum  of 

•e   are  >/  —  1   of 
linal  result  is 


CEShJlM.    THEORY    OF   Eij CATION:'. 


451 


>r  (oeflieieuts  of 

+  ....+0'-2). 

to  form  the  eo- 
)•  -  4,  I +:]  +  (; 
.line  as  these  of 

roenicieuls  are 
ng  table,  where 
-  the  one  ou  its 


,fl 

r 

r= 

r* 

jA 

/-'• 

;•*■'     etc 

/<o 

1 

1 

1 

1 

1 

1      etc 

h 

•> 

V 

>> 

4 

5 

r. 

etc. 

/<2 

;{ 

<; 

](► 

15 

etc. 

A3 

4 

10 

•^0 

etc. 

A< 

5 

15 

etc. 

A» 

0 

ete. 

¥ 

etc. 

etc. 

etc. 

u 


u 
u 
a 
<• 


left.  We  have  carried  the  tahle  as  fur  as  n  =  C,  jukI  the  ex- 
pressions at  the  h(»ttom  of  each  column  will,  when  it  ■=  (I,  he 
formed  from  the  numhers  in  this  table,  taken  in  reverse  order, 
thus : 

Column  under  i/,     i'xtr  -f     /»; 

•'      r,  15r//''-l-   bbr  4-   r; 

(I,  2(  V/r^  f  id///'-  4-  irr  -\-    >i  : 

e,  15r?H4-H>/»/'3-f-(;/v-4-:)r/r  +  p\ 

/,     VHir'-\-  5/vrH  4r;-^+:i</?-^-f  :i/r4-/; 

/7,       ^^/*4-     ///•=^+   rH-i-   ^/r^4-   i'r~-{-fr-\-(/. 

\ow  the  i.uml)ers  of  the  above  scheme  are  the  tiiiurate 
numlRTs  treat<-tl  in  §  2^1!,  where  it  is  shown  that  the  y/"'  num- 
ber in  the  <'*  eolumu  after  the  column  ornnit>  is 

njti  +  \){n  +  'i) (w  4-  /  —  1 ) 

l-i.3 /  "        \         t 

Conijiarin^-  with  the  coellicients  in  the  ef|uatM)ns  (.(),  wr 
see  that  the  two  are  identical,  which  proves  the  correetnes;  rf 
the  method. 

370,  Ajiplirntion  of  tJic  Prccrdinn  Ojirrafion  ta  ihf  K.r- 
tracti'jit  iff  the  linots  of  Numrrical  K(jtta(io/tf<.  Let  the  equa- 
tion whose  root  is  to  ])e  found  be 


-•  =  CL±^). 


a.c"  4-  //.r"-'  -f  f.f*  -  4- 


^-    f/   r=    0. 


We  llnd.  by  trial  or  otherwise,  the  frreatest  whole  nnntber 
in  the  root  x.     Lei  r  be  this  number.     \Vc  sul)st?tute  r-^/t  for 


4r^'2 


({!:m:um.   riii:(n:y  or  i:\>r.\ri(>ss. 


til 


M 


./   ill  iIk'  iihovc  expression,  an:!,  I»y  the  prccrd in*,' process,  get 
an  I'lpiution  in  //,  wliicli  wc  may  pul  in  tlic  form 

ah''  -f  ////«  >  +  r7/'»  ••*  +  (lit''  ^  -f- +  //'  =  ^>- 

Let  ;•'  l)u  the  lirst  deeinial  of  h.  We  put  r'  +  /i'  lor//  in 
this  e(|uation,  luul,  hy  repeat  in;;  the  proeess,  get  an  etpiatiou 
to  (Icti  inline  h\  whieli  will  he  less  than  0.1.  If  r"  he  the 
greatest  nuniher  (»f  hundredths  in  // ,  we  put  h'  =  r" -\-h'i  and 
thus  get  an  e(iuation  for  the  thousandths,  ete. 

J571.  The  tlrst  operation  is  to  liiul  the  numheraml  approx- 
imate \alues  of  the  real  roots.  There  are  several  ways  of  doing 
tiiis,  among  whieii  Shirin's  Theiircin  is  the  most  eelehrated, 
Itui  all  are  so  lahorious  in  appliealion  that  in  ordinary  eases  it 
will  he  found  easiest  to  proeeed  hy  trial,  snhstituting  all  entires 
numhcrs  for  a*  in  the  ecpiation.  until  we  tind  two  eonsi'cutive 
nnmhrrs  hetween  which  one  or  ni<)re  roots  must  lie,  and  in 
dilHeult  eases  plotting  the  results  hy  i;  :Vtr>. 

It  is,  however,  necessary  to  he  ahle  to  set  some  limits  he- 
twcen  which  the  roots  must  he  founil,  .uul  this  mav  be  done 
hy  the  following  rules: 

I.  .  ///  ('(Illation  ill  u'hiclh  all  thr  rocf/lcii  !>fs,  iiiclndinij 
tJir  (ihst)Iiif('  Icrni,  nrr  fxtsifirr,  cdii  have  no  po^iitirc  real 
root. 

t\)r  no  sum  of  positive  (|uantities  can  he  zero. 

II.  //'  ///  coui patiii'J  the  r/itnr  of  Fx  for  an  if  (issmncft 
positirc  riilKr  of  x,  Inj  t/ir  prormfi  of  j5  ;j(i(J,  ice  find  (ill  tlie 
Slims  j)osilirt\  there,  eaii  tte  no  root  so  ^rent  as  that 
nssuined. 

For  the  snl)stitutioii  of  any  greater  nuniher  will  make  all 
the  sums  still  greater,  and  so  will  carry  the  last  sum,  (ir  l\r, 
still  further  from  zero. 

III.  If  the  Slims  are  alternatelii  jmsitive  and  nvja- 
tive,  the  rat  lie  of  x  ice  einf)lo!/  is  Jess  titan  ((itij  root, 

IV.  If  two  rallies  ofx^ire  (lijferent  si'Jns  to  Fx,  three 
must  he  one  or  some  odd  nuinher  of  rtKjts  hetireen  these 
values  (compare  §  M-fo). 


I 


I 


1(L\\\ 


ai'jyEiLiL  ruE<niy  of  i:qr.\Ti(>ss. 


453 


(lin<,'  process,  gi't 
nil 

4-  U'  =  0. 

t  r' -\-h'  ibr  //   in 

•jjct  an  i(|nati»)ii 

.1.     ir  /•"  lie  11)0 

'*'  =  /•"  +  /*',  jiiul 

nher  and  ajipro::- 
'I'al  ways  of  (loin-; 
most  ft'k'hratoti, 
ordinary  cases  it 
ilutinf;  all  entire 
[  two  consecutive 
must  lie,  and   in 

t  some  limits  hc- 
tliis  mav  be  done 

,"/.«?,  ill  ('111  dill 'J 
\ii(>  f)usiiii'('  rcdl 


■ro. 

;•  (in  11  (tssNiiK  (f 
[ire  find  (ill  flic 


ijl'l'tlt    (fS 


that 


•r  'wili  make  ui 
ast  sum,  or   /'.r. 


■rr-  and   nc^ii- 
hnii/  mot. 

I//.S'  fo  F.i\  llirrc 
betireeii  tJieso 


V.  Tim  ludara  nf  x  irjiirh  lend  tn  the  same  si'Jii  of  Fx 
iiichidr  riflirr  no  roofs  or  (in  even  number  of  roofs  be- 
fii'ien  til  (III. 

Let  us  take  as  a  first  e-xaniple  the  e(|uatioii 

a:3  _  ;j.  _^  7  _  0. 

Let  us  Hrst  assume  x  =  1.     We  compute  as  follows  : 

(VK-nieic'iits,        1       0       —  r       +7 

Products,  _4  Hi  'M 

Sums,  +4  +r>         +43 

So  F(4)r=  +4:},  and  as  all  tlu'  ('oi'Ulcii'nts  are  positive, 
there  can  he  no  root  as  <j:ivat  as  4. 

I'uttinj]^  x=  —1,  the  sum>,  iiuludinir  the  tirst  coellicient 
1^  jii-e  1,  —4,  -f-!»,  — '-iO.  Thtse  l)ein<^  alternately  i)ositivv  and 
ncirative,  there  is  no  root  so  small  as  —4. 

Siihstilutin<,'  all  inteprers  hetwcen  — t  and  +1,  we  llnd 

F{-\)  =  --.'O.  /'(O)  :=  +  r, 

F(_;j)  =  +   1,  /'(I)  =  +  1, 

/'(_->)  =  -^1:5,  /'(•-')  =  +  V. 


F{-\)  =  +1: 


F{:i)  =  +13. 


we 


If  we  draw  tlu*  curve  corresponding!^  to  these  values  (J^  34')), 
shall  lind  one  ro(»t  hetwi'in  —3  and  —4,  ami  very  near 
— 3.0."»,  and  the  curve  will  di[)l)('low  the  base  line  hetween  +  I 
and  +'■!,  showing  that  there  are  two  roots  hetween  these  num- 
that  is,  there  are  two  roots  of  the  form   1+//,  //  hein.u^  a 


iK'rs 


positive   fraction.     Transforming;  the  e(|uation    to   one  in  //, 
l)V  ])utting  I  +  A  for  ./■,  we  tind  the  eijuation  in  //  to  he 

7^3  _|.  15/^2  _  4//   f  1  =  0.  (!) 


one 


Suhstitutini'  /^  —  0.v\  <».4,  0.0,  O.S,  we  find   that  there  i^ 
root  between  0.3  and  0.4,  and   om-  between  0.(5  and  0.1. 


Let  us  begin  with  the  latter. 

If  in  the  last  equation  we  jmt  // =  0.r,  +  //',  we  find  the 
transformed  equation  in  //  to  be 

Fit'  =  Jr^  +  4.S//''  +  O.OS//'  —  0.104  —  0.  (^^ 

If  we  sulistitute  dilferent  values  nf  //'  in  this  e(juation,  we 


454 


a/'jNh'itAL   TJihoiii'  OF  hyr.iT/oys. 


ii 


'» 


shall  liiid  that  it  iiiiist  exceed  .00.  and  as  it  nuisl  he  less  than 
0.1,  we  eoiiclude  that  1)  is  the  li«fiire  sou^Mit,  and  put 

//•  =  .Oi)  4-  //■. 

Tninsi forming.,'  the  efjuatiun  {'i),  we  lind  llu'  equation  in  h' 

to  bo 

// "3  4-  5.07 h'^  -\-  1.5(18:}//  '  —  o.oo.'U'jl  =  u.  (:j) 

Since//'  is  iicei'ssarily  less  tha:i  D.ol,  its  lirst  di<;it,  whitu 
is  all  Wf  want,  is  easily  found,  heeause  the  two  th'st  terms  oi 
the  ei|uatit>n  are  very  snudi  compared  with  the  third.  So  we 
siinidy  divide  .('0:n01  hy  l.oOS;},  and  lind  that  Am  is  the  re- 
quired diL'it  of  //".     We  now  put 

h'  =  .()()•»  +  /' ", 

and  transf»)rm  again.     The  resuUin«,M'i|uation  for//"'  is 

//"«  4-  .').<):(;/<  "••2  +  i.r)Ssr)W.7/"  —  o.(;()(K);mii-^>  ~  o.    (i) 

The  diijits  ol'  r,  //.  //',  and  //"  which  we  have  found  show 
the  true  value  ol'x'  to  he 

By  continuiiiir  this  j^roccss,  as  many  liiruros  as  wc  please 
may  he  loiiiul.  i5ut,  after  a  certain  point,  the  opcralion  may 
he  althrcviated  hy  cuttin<j;  oil'  the  la-t  ligures  in  the  coeHieicnts 
of  tlie  powers  of  //. 

The  work,  so  far  as  wc  have  performed  it,  may  be  arran-eil 
in  thi'  foiiowini;  form  (see  next  pa<i:c). 

Till'  numbers  under  the  double  lines  arc  the  coenicicnts  of 
tlie  powers  of  //,  //',  h'\  etc.  It  will  be  seen  that  for  each  di-,Mt 
W(,'  add  to  the  root,  we  add  one  di<(it  to  the  coellicient  of  //', 
two  to  that  of  //,  and  three  to  the  a])solute  ti-rm.  We  have 
thus  extended  the  latter  to  niiu'  places  (d*  decinuds.  which,  in 
most  case's,  will  give  nine  lii:ures  of  the  root  correctly,  if  this 
is  all  we  lU'ed,  we  add  no  more  decimals,  but  cut  off  oiu^  from 
the  coi'tliejeiit  of  //,  two  from  that  of  //'.and  soon  for  cacli 
decimal  we  add  to  the  root. 

We  shall  lind  the  next  figure  after  X.Cd'i  to  1)C  zero  ;  so  wo 
cut  olV  the  llirures  without  making  anv  change  in  the  coclli- 
cients.  The  next  following  is  'i,  so  we  cut  ofl' again  foi-  it.  and 
multiply  as  shown  in  Hie  fnll'.v.-ingcoiitinuatifui  of  t hi' process: 


loys. 

list  Itf  less  tliuii 
iiiJ  put 

L'  L'(|ii;itioii  In  h" 


1   ==  0. 


(a) 

iitit  ili^it,  wliicu 
v<>  tinst  k'nns  oi 
K'  third.  So  we 
t  .OOv*  is  the  re- 


or  //     IS 
{4ll;>=:(>.     (I) 

luve  fouml  show 


'08  SIS  we  pleiiso 
?  operation  niny 
1  the  eoetlicicnts 

nay  he  arran-vil 

■  eoellicicnts  of 
ii  lor  eaeh  tliuMt 
ni'lheienl  oi'  /r, 
t'vm.  We  liave 
inals,  wliieli.  in 
•rcctly.     If  til  is 

It  oft'  one  from 
.-^'o  on  for  eacli 

i)C  zero  ;  so  wo 
:c  in  tlie  eoelli- 
p:ain  for  it.  and 

of  tlic  pr(»eess: 


VLWJJIiAL    TIIKORY  OF   EqCATInXd. 


0 

+  1 

+  1 
+  1 

+3 
+  1_ 

+8:0 
±  -^ 

_J 
4.9 

6 

+  4.M0 

9^ 

4.M1» 

y 

4.JW 
9 


•i.'w 


-7 

+  7  '    XXAfU 

+t 

-« 

^ 

+  1  IMHI 

+2_ 

-1.11)4 

-4.00 

— ".I04t»>o 

+  .HM)H09 

-184 

+  o.«;h()0 

-  .no;!l!»Hi»i») 

KUIOl 

+  1.1L'()| 

+  .4  ISO 

+  l.r)(iH:!()n 

l(tl44 

Vl.r»TM4t4 

10!  18 

4  i.r)8>ir)i»-> 


+  .').070 
o 

~r»7of3 
3 

6.074 
+  5.076 


+ 15.070 


CONTINU.VTION   or   PTIOCESS. 

4l.r)SS'>S)2 
1 

i.r.ss? 
1 

1.5|yi8i8 


:{IT;4 

-   74'.» 

—11:5 
111 


If  will  !x?  scon  that,  from  this  judiit  we  make  no  ii5*>  of  ilio 
coelliciiiit  1  of //3.  and  only  witli  tlie  seeond  dceimal  do  we  i\st.* 
the  coelHeient  of  //-.  After  that,  the  nmaining  four  tlir-ires 
are  olitaim-d  hy  pure  division. 

There  13  one  thinir.  howevor,  wliicli  a  eonipnter  .*ho:!«Kl 
always  ;!ttond  to  in  nmhipiyin^  a  niimher  fr.)Ui  whieh  lin.'  \ias 
cut  olTtiirures  in  this  way,  namely: 

.Ihi'rnfsi  rnrnifo  Ihr  jinxlucl  Uic  iinnihrr  irJiirh  iif,iihj 
have  been  currictl  if  (he  jl'JiiretihtKi  not  ht  en  rut  njj'.  antl 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


//J 


// 


{/ 


,% 


^A 


fc'  #^< 


/ 


i/j. 


(/. 


1.0 


I.I 


11.25 


f.  iM  IIM 

I  i  Ilia 
I  .^  Ilia 

1118 


iA  III  1.6 


^ 


& 


/a 


/ 


'c5. 


Photographic 

Sciences 
Corporation 


23  WEST  MAIN  STREET 

WEBSTER,  NY.  14580 

(716)  872-4503 


%^ 


iV 


iV 


\\ 


<\ 


'^.1^ 


'b''  .A  ^ 


rt^ 


r^^ 


4,56 


GENERAL    THEORY  OF  EQUATIONS. 


Il 


M 


m  i  I 


increase  it  hy  1  /'/  tlic  figure  following  the  one  carried 
u'oidd  ha.ve  been  o  or  greater. 

For  instuncc,  we  had  to  multiply  by  7  the  inim])cr  lo'888. 
If  we  entirely  omit  the  figures  cut  off,  the  result  would  be  105. 
But  tlie  correct  result  is  llli^lG;  we  therefore  take  111  in- 
stead of  105. 

Again,  in  the  operation  preceding,  we  had  to  multiply 
158i88  by  4.  The  true  product  is  G35,52.  But,  instead  of 
using  the  figures  G35,  we  use  G3G,  because  the  former  is  too 
small  by  |52,  and  the  latter  too  great  by  |48,  and  therefore  the 
nearer  the  truth.  For  the  same  reason,  in  multiplying  1.58818 
by  1,  we  called  the  result  1589. 

Joining  all  the  figures  computed,  we  find  the  root  sought 
to  be  1.G92021471. 

Let  us  now  find  the  negative  root,  which  we  have  found  to 
lie  between  —3  and  —4.  Owing  to  the  inconvenience  of 
using  negative  digits,  and  thus  having  to  change  the  sign  of 
every  number  we  multii>ly,  we  transform  the  equation  into  one 
having  an  e([ual  positive  root  by  changing  the  signs  of  the 
alternate  terms.     The  equation  then  is  a.*^  —  Ix  —  7  z=  0. 

The  work,  so  far  as  it  is  necessary  to  carry  it,  is  now  ar- 
ranged as  follows : 


0 
8 

8 
8 

6 

3_ 

9.00 
4 

9X)4 
_4 

9.08 
A_ 

9.120 

8 

9.128 

H 


-7 

2 

18 

2070000 

20.3(JIG 
.3632 

'20.724800 
J'3024 

20.707824 
73088 


-7  13.0489173395 
6 

-1.000000 
814404 

-0.18003(5000 

.1()G:582592 


20.8709112 
^823^ 

2a879l'4;3 
823 


0.136 
^ 

|9.1|44 


20. 8^73 1 7 
9 

201.8  8i7|5 


.19153408 
1S791228 

-362180 

_208875 

-153305 
146213 

-7092 
_6266 

-826 
627 

-199 

188 

-11 


TIONS. 

<  the  one  carried 

10  number  15'888. 
suit  would  be  105. 
•efore  take  111  iu- 

had  to   multiply 

But,  instead  of 

!  the  foi'mer  is  too 

and  therefore  the 
iiultiplying  1.58818 

lid  the  root  sought 

I  we  have  found  to 
inconvenience  of 
3hange  the  sign  of 
e  equation  into  one 
Of  the  si^ns  of  the 
.  7a:  —  7  =  0. 
arry  it,  is  now  ar- 

!  3.0489173395 


1000000 

I8144G4 

ls:iT8()000 

|UiG:382r)93 

19158408 
1S791228 

-8(i2t80 

208875 

-158305 
14(3313 

-7093 
J)266 

-836 
637 

-199 

188 

-11 


GENERAL    THEORY   OF   EQUATIOXS. 

The  negaiflvc  root  of  the  equation  is  thei-efore 

—  3.04891:3395. 

EXERCISES. 

Find  the  roots  of  the  following  equations: 

1.  x^  —  3x^  +  1  =  0  (3  real  roots). 

2.  a-3  —  3a;  +  1  =  0  (3  real  roots). 

3.  X*  —  4x^  +  2  =  0  {'Z  positive  roots). 


457 


4. 

5- 


.r2  +  X 


0. 


Prove  that  when  we  cliange  the  algebraic  signs  of  the 
alternate  coefficients  of  an  equation,  the  sign  of  the  root  will 
be  changed. 

37!3.  The  preceding  method  may  be  applied  without 
change  10  the  solution  of  numerical  quadratic  equations,  and 
to  the  extraction  of  square  and  cube  roots.  In  fact,  the  square 
root  of  a  ntimber  n  \s  a  root  of  the  equation  x^  —  n  =  0,  or 
x^  -f-  Ox  —  n  =  0,  and  the  cube  root  is  a  root  of  the  equation 
xi  -f.  0.^2  +  Ox  —  n  =  0. 


Ex.  I.  To  compute  V2. 


0 


9.0 
04 

2.4 


2.80 


2.81 


-2  I  1.4142ia56 


-1.00 


.96 

-.(MOO 
381 

-11900 
11290 


3.836 
4 

2.834 
4 

278380 
3 

3.8383 
2 

-60400 
56.564 

2828 

-1008 
_849 

-1.59 
J^41 

-18 
J7 


2i.8:2  8  4 


458 


GENERAL    THEORT   OF  K'^UAllONa. 


Ex.  2.  To  compute  the  cube  root  of  08-t20;3(v 


0 

0 

2 

4 

a 

4 

2 

8 

3 

1200 

d 

61 

60 

1261 

1 

62 

61 

132300 

1 

2r)3G 

62 

18'4836 

1 

2552 

030 

137388,00 

4 

192.09 

634 

137580.69 

4 

192.78 

638 
4 

13777377 

1.93 

642;0 
.3 

1317|7|75i4 

(543.3 

3 

643.(; 

3 

642.9 


-9842036  1  214.30300243 
_^ '  ' 

-1842 
1261 


-581036 
539344 

410!»2000 
412742(J7 

-417793 
413326 

'  4167 
4i;i3 

"334 
276 

68 
55 

8 


I 


'I 


M^ 


^42036  I  214.30303243 


ANSWERS. 


58 


TN  the  following  list,  answers  to  questions  which  do  not  require  cal- 
X     culat.on  or  written  work,  or  which  It  is  supposed  teachers  would 
pre  er  to  have  in  a  separate  Key,    are  omitted.      The  Key,    published 
lor  the  use  of  teachers,  contains  the  complete  solutions. 

26.      I.   -9.     2.   -17.     3.   +9.     4.  -26.     5.   -f  10. 

6.  -15.     7.   _5G.     9.   +840      10.  -105G.     11.    +1. 
12.   —306.     13.  0.     14.    —1008. 

38.      I.  1.     2.-2.    3.-5.     4.-14.    5.  +24.     6.?^  =  ?. 
2G      „    1  9        3 

40.      I.  0.     2.  0.     3.  11.    4.  17.     5.  _37.     6.  -90. 

7.  324.     8.  0.     9.    —CO.     10.  —180.     11.  945. 

12.  5040.     13.  -41.     14.  -1.     15.  _i7.     16.  2G. 
17.  99.     18.  G75.     19.  74.     20.  -4G8.     21.   -218. 
22.   -529.    23.  —9007.     24.  —6800.    25.   —420. 
26.    -840.     .7.  ~.     28.  -iL.     29.  2.     30.  8., 
31.  When  X  =  2,  Exp.  =  6;  x  =  5,  Exp.  =  18 ;  .-c  =  7, 
Exp.  :..  36.    32.  When  x  =  -5,  Exp.  =  -  ^  ;  ^r  =  2, 

Exp.  =A;.^:=5,  Exp.  =  -1. 

43.    I.  When  x  =  -3,  Exp.  =  0;  n-  =  -  1,  Exp.  =0; 

^  =  1'  E^^P'  =1 ;  a:  =  3,  Exp.  =  15.     2.  When  x  =  -3, 
Exp.  = --;  .  ^  _1,  Exp.  =  |;  .  =  1,   Exp.  =  1|; 
a-  =  3,  Exp.  =  24.     3.  When  x=  -  3,  Exp.  =  46875 ; 

a;  = -1,  Exp.  =  —  - ;  cc  =  1,  Exp.  = —88434 ; 

cr  =  3,  Exp.  =  -  ^  (365)3.     4.  When  x  =  -T,  Exp.  = 

(^14-^/2)4;    .'r  =  l,    Exp.  =  (^8  _  V^)  4  ; 
«  =  3,    Exp.  =  (^48  -  V42)4. 


4no 
48. 


54. 


55. 


56. 


ANSWERS. 

I.  a  4-  ia:  —  (.t;  —  y).     2.  .'?;  —  2/  —  («  +  ia:). 

a  —  bx  a  —  hx  , r- 

4.  — ;;7^ w/;?*/.     5.   va  +  bx. 


3.  rt  -I-  bx  — 


wz 


«i 


6.   A/(rt  4-  bx)  ■\-  {x  —  y).     7.   \/(rt  +  bx)  —  {x  —  y). 
8.   {a-\-bxY{x  —  yy,     9.   {mpq^.     10.  {x  —  yY  {mpq)'^. 

{a  —  bx)  {x  —  y) 
m 


mpq  {a  +  bx)  — 


II. 


r-^r-(^-^)' 


etc.,        etc.,        etc. 
I.  ha  +  4J  —  8c  —  <?.     2.   —  «  +  (a;  4-  y).     3.  G. 
4.  Six  —  13y.     5.  23  («  +  J)2  —  a;  —  2/  —  z.     6.  5  (aJ). 
7.  0.    8.  't!{m-\-nY—x—2y.    9.  4(;j  +  ^)2  +  «  +  ^»4-c— 6. 
10.  14a  {x  —  y).     II.  15  {}n-\-n)  x  -{-  2  (in  —  n)  x  —  17. 


12.  7-4-31-1.     13.  10- -10-.     14.  IG 

(lb  y        w 


a;  4-?/ 


m  4-  vi 

15.  5a:  —  '7y.     16.  8;r.     17.  4a;  —  30. 

I.  {a-^7n)x  +  {b-^n)y.     2.  {mn -\- pq)x -\-  {2b  —  ^h)y. 

3.  (3  4-  G^>  -f  7«)  a:  4-  (—  3  —  4)  ?/  4-  w  4-  n. 

4.  (Srt  4-  8J  4-  7  4-  1)  a;  +  (^  —  5  —  5)  y, 

5.  (rt  —  w?)  X  -\-  {b  —  7i)  y  -j-  {c  —  p)  z. 

6.  (2(1  —  2f)  X  4-  (3e  -  3fZ)  ?/  4-  (4/  +  4e)  0. 


a:. 


8.  {2a  —  3b)  X  +  {  —  b  —  4d)  y. 

a  1    \  /3  ,       3 

9.  (^a_-.,)a:+(,i  +  - 


3    \ 


10. 


'^K  If- 
y-m  —  3«  —  Gc  +  hi)  X  +  {2 


^-ci)y. 


II.  {bab  —  ab  —  d)x  -\-  {iccl  —  3mn) y. 


12. 


(-.-1.) 


a;  4-  5ay.      13. 


-8a;  +  (3-^«)y. 


14. 


(3w  +  1  4-  rt  —  rt)  a:  4-  (—  1  —  2  «)  2/- 

15.  3abx  +  (2c  4-  1)  ^/x  4-  (—  ?r.  —  a)y. 

16.  —  Ga;  +  (5772  4-  5)  '\/y  —  y  —  %^x. 

17.  ex  4-  h^/x  —  Gy  4-  (—  3rt  —  1)  Vy. 

3.   -  llrt  4-  lO^*  -  4c  4-  7^/  —  7a:  4-  (4  4-  3c)  y. 


\ 


ANSWERS. 


4  til 


I  4-  hx). 

pq.     5-   Vd  +  t'-f- 

~bx)  -{x-  y). 
{x  -  yf  {inpqy\ 


-f  y)'    3-  c. 
y  —  z.     6.  5  (ab). 
qY  +  a  +  b  +  c—e. 
I  {771  —  n)x  —  17. 

14.    10 -— ;• 

)x+  {2b-'ib)y. 
m  -\-  n. 

5)7- 

2:. 

4:6)0. 


a)2^- 


y- 


4  4-  3^)  y. 


% 


69. 


4. 
6. 

8. 

10. 

12. 

14. 

15- 


11 7^  +  283^2  +  ^'iy  —  57rfa;  -  20.     5.  2rt  -  (;/>. 
2«  —  Hb  -\-  2c  —  2(1.     7.  4a  +  U  -[•  ic  -\-  2d. 
—  ;U-^  —  2a;  —  4.     9.  32-4  —  ar^  +  14a;  -\-  18. 
a;a  —  ^a;  +  2«l     n.  2a^  —  6a^  +  3rt/!;5  _  ^,3, 
3^  4  4a;  +  lO.     13.  —  4  (a;  —  //)  +  '4  {z  —  x). 
o  {a  —  b)  -}-  2  {a  -\-  b)  +  7a  —  26. 

12-_17-^-8^_8?. 
y  z         x         b 


58.      I.  2a-.     2.  2^.     3.  4rtZ> 
-  3rti  —  ?/i  —  2rta;. 


59.     I. 

4- 

61.     I. 

3- 
6. 

10. 

12. 

13- 

15- 
2, 

6. 

10. 


■imp—Sx.    4. 
2.  3a-  —  2flr. 


mx—pz. 
3- 


5.  .'i 


a 


13- 
70.    15 

19. 

22. 

73.      I. 

5- 
8. 

II. 

14. 

17. 

20. 

23. 
26. 

29. 


2* -4c. 
lOx—7y-\-i)Z.     5.  —dax—2by.     6.  0.     7.  0.    8.  3//j. 

7n—2)-\-q-}-a—b  +  c  +  (l    2.  m  4.^— i4-/>4-<7— ;^4-^'. 

ISrt.t  —  4:by.     4.  0.     5.  y;  4-  6  4-  5  4-  ^  4-  m  4-  n. 

llax.    7.  —2ax—(Jby—cz.     8.  —2a; +  2?/.     9.  —  4iz. 

2a;  —  Gy      my  +  4rt?>  —  5.     11.  «a;  4-  2cx. 

'iax  —  35.7;  4-  ^ay  4-  3«0  —  Zby  —  3bz. 

13ax  —  '3xy  —  2d  —  7ftr/.     14.  in  +  dx-\-4y—ay —  p. 

2aA/y  4-  Vy  —  3y^j  4-  G;?  —  b's/x. 

G(f%r^.     3.  15miry.     4.  42ah)i^y.     5.   4fl!2/w2^ 

5a;y«2.     7.  O.cY^l     8.   4«2Z>2;;,2.     9.  Qa^bix*. 

14:4 ))ip\fr''^s.     II.  144«a;2^2z.     12.  7)i^\v^y^. 

3mn^k^.     14.   \4((bcd\ffj. 

m^xyz.     16.  abcdx^.     17.   12aW-m^n\     18.  14a^b^c^. 


21.  a''nrn\iyy^z 


l^iitn^n^p^.  20.   GaHmlmifz^. 

a^^x~y\     23.  48«4/;?^;<2a-2. 

a'^bcdm.     2.  —abcdxK     3.    —a^b'^cx^.     4.  30rt''J'';/«a-l 
lOoa^m^xy^.     6.  lOwV^^'^y/^l     7.  4abmn. 
IGSnhn^kx^     9.   Obnuu/y^.     10.  4((x^y'''. 
—30af/x^yh^.     12.  Ibayfhix'^yz.     13.   —4abyxyz\ 

4bc^gn.rh\  15.    —3al^e^x^y.     16.  4abcxy. 

—24(thhf.  18.  «'.r?;/3,     i(j.   _3a^.?;3v3. 

—  nf^hh^.     21.  a^bx^i/^.     22.   —apqxhf. 

2 

24.   dacm'\iih'\     25.   —  ^acmhi^x\ 


da^cxy\ 
vn^ii^x^y. 


5 


27. 


—a%dx\     28.   —  30rt2wj4^^4^, 
30.    —  -  m^pqx^y^ 


r 


403 


ANSWEIiS. 


1^* 


*'** 
i*i. 


'» 


74. 


7G. 


78. 


71). 


2. 

4. 
5- 
7. 
I. 

2. 

3- 

4. 

S- 
6. 

7- 
8. 

9- 
I. 

2. 

3- 
4- 
I. 

2. 

3- 

4- 
5- 
I. 

3- 
4. 
6. 
8. 
II. 

12. 

13- 
14. 

15- 
17. 
18. 

19. 
21. 


a^x^i/z  4-  (i/jxi/h  -\-  (trxt/z^. 

^7rrV/r*  —  ■ii^><(V)Xif  —  Wonbj'.     6.   —i'iin^j^q  -f  i8>/(  ^^^i^^ 

(tp  -\-  Dtp  —  p^  +  b(i  —  tY/  —  hr  —  cr. 

mx  —  anx  —  nnj  —  any  ■{•  anz  —  viz. 

acx  —  act/  —  bdx  +  bdy  +  fcdx  +  fcdy. 

avix  —  d^bm  4-  dkm  ~  abnx  —  i^^^^  —  bhid. 

—apni—apn  +  Z»;?^>i  —bpu — bip-n  -\-  bqn  -\-  aqni  +  aqn. 

(jqx  —  'Siicx  4-  l().?'y  —  Oci/  —  2;:r;>/  —  72^2. 

fi^mh'  —  mi^bc  ~  iutmhk  4-  Vlainhd  4-  Uimn. 

imp  I  —  IQbpq  —  V2rpq  —  im/i^q  4-  (Jnp'^q^, 

—  7f//>^/  —  7<fbh>  +  7/A'M  —  U)i  +  (dm  +  Uhi.      10.  0, 

(a-2  4-  'Zx))f  4-  (;}.r=J — '>./:2  —  1  4-  5.?-)/y2 — Ax^y  +  a'2 — 7,r — 6, 

.7;'<i^4    _j_    ^^;y3    _j_    (i    _     ,;Jj  yi   _   j-y   1. 

xhf  4-  .'T^'//'  4-  (.-^  —  'ix^)  v/3  -f  (1  —  2a-2)  _?/2  —  2xy  —  2. 
^V  4-  xY  4-  (3.i;3  4--a.-2)  7/3  4-  (3.^44.3)/ 4-  2x^y  +  dx. 
2a:i  —  abii^  —  2((bii^  +  2ub  —  b'^h^  —  W^ifl. 
3(0)1  4-  2aii  —  5d^bmu  —  3^^;/  —  2b)i  4-  babhnn. 
2m^}i  +  pin^  4-  qm'^n  —  2mn^  —  pmu^  4-  </>/''. 
p'^q  4-  ^>2^r  +  ;;'';•  4-  pq^ -\-  qh'  4-  pqh  -\-  pq)'"^  +  (//'^ 4-  2)r\ 
4rt'^  —  Jif/5  —  W.     6.  y??2:c2  ■ 
U^  +  «3  _L  11^,2  —  a^  28. 


2.   rt^  —  J3. 


r/^  4-  rt^  4-  <-(^'^^  —  i'^^x  —  (i^x  —  X*. 

ft5  _  2a*  +  da^  —  3rt2  +  2ft  —  1.     5.  x'^  —  rr'. 

am  +  im-^cmz^  +  dmz\    7.  C«*  +  19rt3+17«2  +  r<— 28. 


r«3  4-  &l     9. 


rt"  —  a-i. 


10.  «5  _.  ^3  _|_  ^2  _  2«  4-  1. 


Q^  4-  2«a;4  4-  2«2^^  +  2a^}?  -\-  2aiv  +  a\ 
am  4-  («;i  4-  bm)  z  4-  (iw  4-  e?)i  —  ap)  z^ 

4-  {dm  4-  c?i  —  bj))  z^  4-  (<^Zm  —  cp)  z*  —  f//;^^ 
«;»  +  {an  4-  Jm)  .r  4-  hu^. 

am  4  ('^^^^  +  bm)x  +  (^7;  +  bn  +  cm  )x^  +  {bp  4-  cn)x^  +  r;):^*, 
if  —  by^  -\-  2f  4-0^  —  4.  1 6.  ?/5  4-  2^  4-  3_y3  .y  if  4- 1. 
y6  +  2?/*  —  7/  —  10. 

(3r<3m _ 3«?ffi+« ).j.  ^  (  _ 3f,m+3  ^  3f^;i4 2)  y  ^  2«"*+2"  - 2ft3«. 

1 


17  1 


2«i2 


Z'l     20.  4«Z». 


a^ 4- 2^3 4- a2  _  j4  _  o^,3_^3_     22.  ^2  _|.  o^^^,  ^  ^.2  _  ja. 


1  f 


cr. 

VIZ. 

-Ml/' 
'ic  —  d^nd. 

bqu+iujin+aqn. 

-  7zn. 

I  +  iamn. 

\-  i\nphf. 

altn-\-b^n.      to.  0, 

-l-c-^TZ  +  a^— 7?^— 0, 

1. 

2a;2)  ?/2  —  2xij  —  2. 

—  -Zbhi^ 

p)i  +  bahhnn. 

nn^  +  qti^. 

-{-  pqr^  +  qr'^+  p?'^. 


5-  ^^  — 

a^ 

)^/3+17«H^« 

—28. 

4-  rt2_ 

2a  +  1. 

f  «'\ 

ft/;)  ^2 

'  -  n>) 

z^- 

f/?^^^ 

{bp  +  oAx^-^cpx^. 

y  ■}-2a"'^-'' -2a^^. 
JO.   iftZ*. 


\ 


i 


.4i^STVA'i?.S. 


j»;;j 


22. 
24. 

25- 
26. 

90.     I. 


3- 

7- 

9- 
II. 

14. 

15- 

18. 

20. 

I. 

3- 

5- 

7- 

9- 
II. 

104.  7. 

105.  I. 
lOG.   I. 

6. 
10. 


^2  4-  2ac  +  ^-2  —  J^.     23.    -  8^2*. 

_  x2  4.  (;j^  _  ft)  ,c  j^if+{b  —  3ft)  y  +  2ft2  —  ':bK 


r»//i 


97. 


2 

Q.,  a;  —  3  H •     2.  ;r2  4-  3.^  +  1. 

a;  +  1 

^2  +  ft  —  1.     8.  Q.,  x—l-\ --' 

;c  +  1 

4ft-  —  10ft  +  25.     10.  ft*  —  ft'^  +  ft2  _  rt  -f-  1. 

ft2_:^«-j-3.     12.  fl3— 2ft2-|-2«_l.     £3.  a:4_io^_|.iG. 

900 
Q.,  .r'  +  2.c2  _  15a;  +  50  -  —  -- • 

a:  -}-  4 

1  4-  2a;  f  a;2.     16.   1  —  .3a;  4-  x^.     17.  3  —  2a  4-  ft2. 
1  _  2y  4.  2?/2  —  ?/3.     19.   —  IG  4   8.^— 4a:2  4-2.63— a:*. 

Q.,  10  4-  10.^  4-  8^  +  4.r«  +  2^-*  +  7-^— r-2- 

4  4:X  -{-  x^ 

a;2  —  (ft  4-  c)  X  4-  ftc.     2.  a;2  —  (ft  4-  Z»)  a:  4-  nb. 

ft2  4-  (W  4-  6'2  —  fti»  4-  Z»2  4-  be.       4.    ft2  _j_  rt  _  rt^,  4-  ^^  4-  J  4-  1. 

ftj  4-  bx  —  ax.     6.  ft*  —  4ft2^c  4-  7Z»26'2. 

ab  i-  ac  +  c^  -\-  be.     S.  c  -^  b  —  a. 

ft2  _  ab  4-  J2  _  ^<c  —  ic  4-  c2.     10.  a:2  4-  2ft-a;  4-  2ft2, 

ab  4-  ax  —  bx.     12.  x  —  b.     13.  Qah-^  —  ■\:a\i?  -\-  a*. 

m  —  71       m  +  n 

a  —  b        a  +  b 

l-b  +  bc 


be 

X 

«2  -  U^' 


2. 


2. 


X 


a-b 


4-  r     *'*  i~a;2" 


«4- 

ft  4-^ 
, •    5. 

fta; 


2a: 
, -,-     4- 


ft  —  h  —  r  -\-  d 
4.   _„ 


o 
O 


a;(4;c2  — 1)' 


a 


a;^ 


0. 


5.  0. 


ab  +  be  4-  m  —  (ft2  4-  ^2  4.  c2) 


(rt  —  Jj){J,  —  c)  {c  —  ft) 


II. 


2fta; 
x^  —  y^ 


12.  - 

ft2 


4ft5 


2ft 


Z/2 


13-    — 


ft 


+'^' 


14. 


15- 


2b 


X?  (a;2  —  1) 


a  —  b 


16. 


2  (wa:  4-  my) 


{m  —  7i)  {x  4-  y) 


7- 


?/ 


2  —  m^ 


vi^  {m  —  y) 


I     tt 


464 


AiVti  WERH. 


I.S.  ■-^,.- 


21. 

27. 

30- 

107.  I. 

3- 

108.  I. 

14. 

110.  I. 

6. 

9- 
12. 

14. 

111.  I. 

r^3.  I. 

4. 

6. 

8. 
10. 
12. 
14. 


.«^  — ^r'    ' 
'i  {(IX  —  my  +  xj/)  a'i  4.  J3  ^  c» 


ro.  0.     20.       -, —  • 
0 


x^ 


.r 


abc 

0.     24.     o-— TT-    25.  U.     26.  -» -. 

x'^  —  1        ^  x^  —  a^ 


2X1, 
x^  +  y'^' 

\ab 


28. 


—  («  —  .yJl+if 


^9-     (rt3_62)?' 


/I       1       1\  .  pu   ,  pit 

V6'      a      hi  n        m 

I     ,     x/1    ,    1\  .^•  — 4?/       J(w  4-  3a-) 


2m 


2am 


(lb  +  y.     12.  — V- .•     13.  rt6  -I-  2"'  —  a:  I — I-  r  I' 


a 


d' 


b' 


2a  +  3m 


I/  + 


2. 


ax 


y-x 

2x         ' '  III  {an^  —  b) 


u  (am^  +  b) 

7.   ^ :r ,s  •      o. 


ak 
2.ry  -  3 


5-  n. 


{<(  +  by 

2  {a^ -\- 2ab —  d^) 
a^  +  b^ 


10.   1.     II. 
13.  1. 


«2(rt2+2)  +  1 


a  (1  4-  «2) 


(•^•'  +  //•')  (^1-  //)^  +  (g--  +  y)  (--^^  +  y^) 

{X   4-  y)2  (.^3  +  ^3)   _  (^.2  +  ^2) 
y.      4. 5.    X  +  \.      6.   'Y—/i. V 

a  —  b  a  bn  {bm  —  an) 

.r— 27  — 0.     2.  7x—5x=2AhO.     3.  fia:4-42:— 3a;=60. 

a;2  4-  ax  =  w^.     5.  aia;  4-  ab'y  4-  7^^^  =  c. 

5  (4«  +  3J)  =  13.r.     7.  a;2  —  a^  =  2ff2r. 

X  -{-  b  =  2x  -~  2a.     9.  X  -\^  a  =z  '.r?  -{-  2ax. 

a-2  4-3.r— 10  =  a;2  —  3x'  —  10.     1 1.  bx^  —  by^  =  ary. 

x^  —  5a\v  =  0.     i;^.  X  —  y  =:  az  —  bz. 

2x^  ~  ax  —  bx  =  ft2  —  ctb. 


9-    (rt3_^)»' 


—  — —      - —  1 


lb"-      n\ 


-  3m 

_^^n 

ale 

n 

^-3 

b-x) 

n'  +  2)  + 

1 

2  (1  +  «2) 


am  (an  -f-  ^) 
i«  (hm  —  an) 

^  + 4a:— 3a:  =  60. 
2*  =  c. 
ax. 
\-  2ax. 


bz. 


hf  =  a.nj. 


.\XS]Vi:us, 


•!•;.- 


134.    I.  <;;/^-3//-f  4!J  =0.     2.  :}rM-  +  /(8_o.     3.  31^-|-^3  =  U. 

4.  Ha:»  +  'nr/a;3  —  (lrt''./5  4-  a^  —  0. 

5.  <i^/-7/:J  -f  3  (r?3  —  1)  //«  —  ;««//  +  3a»  =  0. 

6.  ^3 .4.  (,,  ^  /,)  ^3  4.  (^,2  ^  ;j^^i  .|.  ^a)  ^  +  ^,a  _|.  ^«  ^  aiji  ^  ^,;!/,  _ ,), 

7.  2r'J  +  rt««  +  ««;?  =  0.     8.   :.//='  +  (i//«  -f  :.^  +  4  =  0. 
9.  .7^  —  r/j3  —  'ZdKc^  —  r/'O;  —  i<«  =  0. 

10.  ;^3  4-  (,b  +  ^O  2^  +  c^;2  +  ^=^  +  *C«  +  c3  =  0. 

11.  ax^  —  ah-  —  b\c  +  b  =  0.     12.   (I— ^yc'  +  w+l  =  0. 

13.  'Za:h*  +  r/x*  —  a^t^  +  ^<3  =  0. 

14.  10-25  ._   13^4  _,_  O^S  .J.  -ll^i  _  (J^  _  3   _   0. 

15.  {a~b)  ,1^  -\-{n-\-b)  x-  +  {d^  —  a^  +  a'^Z'  —  ah)  x  =  0. 

16.  a^T?  4-  (—  «a  +  «'.'^.  _  ab  —  U^)  x  +  a^  —  0. 

33  ^  -k  i  (ihc 

3.   13.     4.   4.     5. 

a  ~  b 

n.    I .       In.    — 

5 

II.  c'-f-rt.     12.  4 J. 

8r«         ,    ^2  (/;  _  a) 


2.    —a. 


ItiQ         T 

3.)  t'r;  +  rt6'  —  «o 

6.   ^.K      7.   3G1|.     8.   '-^ — \.     9.  1.      10.   — '^ 


13. 

17- 

19. 


/>  —  a.     14.  5. 
«  (1  -  /y2) 


15- 


16. 


^i(rt'^-l) 

bn  —  r/>;i 
m  —  n 


18. 


20. 


A  (.^  +  /;) 
a  {((c  -\-  1/  —\)  +  hc^  —  /y 


«  (/^  +  <■)  +  ^c  —  1 
^^3  ^  ^.T  _^  ^3  _  ',l^f^l^ 


d  [c  -  b) 


b  = 


cd 

22.  a  = 7- , 


3  (rt2  +^2  4-  c2 

"rt   +   r/"' 

d)  +  db        .  _         rtJ 
</  '         ^  a  —  b  -\-  c 

cd  ab 


ab  —  ao  —  be) 


h  =  - 


a 


i'r 


d  = 


ab 

0 


130.    I.  20.     2.  72.     3.  I,  $07;  II,  $217.     4-  210.     5.  50. 

6.  180.     7.  05.     8.  A,  i^l30;  B,  *110  ;  C,  $2G0. 

9.  81000,  $1500,  $2000,  $2500,  $3000. 

10.  Man,  36;  Avifu,  30.     11.  I,  18 J  ;  11,  261  ;  III,  45. 

12.  6  ft.     13.  •*2353}4.     14.  81m.     15.   143^8  m. 

16.  A,  $600;  B,  $1200.     17.  8^  m.  per  h. 

19.   15  uiul  24.     20.  15,  10. 


18. 


2  {m  -  h) 


h. 


21.  Man,  40  ;  witV.  35. 


lOi.     23.  CI  days. 


40'^. 


ANSWERS 


2.\.  00  m.     25.  J,  (1;  II, ;{;  JII,  o.     26.  .3000.     27.  100. 
-•.s.   1.    29.  mnni    30.  *l4y.r)0.    31.  I,  *(; ;  11,14. 
jj.  ;{  ni.  un  h.     i^.  $;](;()().     34.  *21800. 

35.  Ji  l>.  ^Vflll. 

w«  ;  IV,  *'*" 


III, 


rt(l  -f  2^^  +  r?2)  '     ''  1  +2,(  +  a^ 


38.  1, ;  JI, — 


Un 


5 


•  III   ^"'  TV  *''  +  ^^*. 


,^   $a  4-  10m 

V, ^ 39-   1^^  li-;  IGO  111. 

131.    40-  30  m.;  3  points. 

41.  JiOm. ;  3  points.     42.  3J- m. 

43-    f^T'T'' 

i;{8.  I.  ^  ==  2|,  X-  =  i'2^.   2.  //  =  7, 0*  ==  in. 

,  7b  m~n  m  \- n 

3.  y  =  a—h,  y  =  -.-  —  a.     4.  ij  =  -^— ,  a^'  =  — ^ — 

5.  /y  =  ^"^--,  •^-  =  '^  .^-^'-     6.  X  =  84,  y  =  84. 

7.  3*  =  33,  ?/  =  50.    8.  X  —  a  -\-  h,  y  =  ^^[a  —  h). 

9.  X  =  1),  ?/  =  3.  10.  X  =  7,  //  =  5. 

It.  y  z=z  (I,  .c  ^  4.  13.  y  =  9,  2;  =  8. 

14.  y  =  8,  .r  =  0.  15.  y  =  0,   x  =  15. 

16.  ij  =  7,  X  =  14.     i-j.  y  =  13,  ;/;  =  0. 

18.   w  = -,  ;i;  =z  — - — ,.     19.   y  =z  2,  X  =  6. 

20.  y  —  r/2  —  Ub  +  U^,  X  =  «2  _|_  ^(^}y  ^  ^^. 

140.    2.  .rj  =  37,  x.^  —  33,  n-g  =  8,  x^  =  V. 

3.  .c  =  3,  y  =  3,  s  =  —  3. 

4.  re  r=  (],  y  =  —  1,  ;2  =  3,  w  =  3. 


5-  ^'  = 


a  -^  b  +  c  —  2d 


y  = 


?^  ■=■ 


a  +  Z>  —  3c  +  r/ 
3 

—  3rt  +  i  +  f  +  (/ 


i 


(;.  .r  =: 


2^  -\-  m  +  « 


.y- 


j/>  +  «  —  m 


z  =■ 


})  —  u  —  m 


i.  3000.     27.  100. 
1,  *tl ;  II,  II. 
)0. 


-3  + a 

tr 


h  a^ 


;  IV,  ^-- 


^a  +  5w 


5       ' 


=  IG. 

/<— w  rn  +  n 

84,  2/  =  84. 


2.  X 


2c  +  ^/ 

b  +  c  -\-  d 


—  n  —  m 


AMSWKIiS.  467 

I.  A,  «i-.>-^r>;  l^*ir)0.    3.  M.    4.  •!•.>.    5.  r,7.    o.  81. 

7.  "^1    8.  A,  80;  13,  ;2.     <;.   KJ  fe'u<)  1,  '^f 5  poor. 
45 

10.'^.   II.  T^,-   12.  •^•),  8.    13.  or,, a;.    14. 5>i,  18. 

15.  A  in  !»,  !iM(l  B  ill  is  d.      lO.  28,  2X     17-  '35,  28. 

18.  1,  40  ;  11,  ;J0. 

19.  B(»n<(lit,  :2'/  and  24if ;  sokl,  iH)^  aud  32^, 

a{h  —  c) 
21.  I,  i;  II,  IJ.     22.  -^J7^— 

23.  A,  .^;]000  (L'^  4,^/;  13,  *4000  (<i)  h%  ;  C,  $4500  d?  0;.'. 

24.  1,120;  11,111;  III,  110. 

KM.  3.  12,  24,  oc.    4.  005,  lo:^.  •'i«f^'  •'^^^4'  :^:^:H- 

i  a  +  2 

(I  4-  />' 


a-b 


8.  a-  = , ,  //  =  .•     lo- 


•     II.  2. 


rt. -2 
14.  *7o;3G.     15.   I,  %57700;  11,  $12000.     16.  8. 

,7.  448  and  1008.     18.  ^  f  ^,  ,7^-y 

19.  :  p.  gold,  5  p.  silver.     20.  5  p.  gold,  3  p.  silver. 


21.    T- 


2nm  +(ni-\-bin 


water ; 


bm+2bn-\-(Ui 


{o-\-b){m-\-n) 


-7 .  alcohol. 


22. 


'Mm^-2an-\-bm  :  ':Sbn-\-2bin -{-an. 


23.  ( ;>  +  q)  am  +  pan  +  qbiii  :  {p  +  q)  bn + ;j6/»  +  qan, 

24.  I,  5  :  3  ;  II,  1  :  3. 

17:5.     T.   1  +  4,c  +  10./:2  ^  i:)^  -I-  9.r». 

2  1  +  4;r  +  lOx-  +  20x^  4-  25.7.^  +  24^^  +  10^,-« 

3  i  +  4./:+10.'kH20./'3  +  25x^  +  34.'c5  +  30.^;6  +  30./-t  +  lO-^"' 
+  25aio.  4.  1+ 4:/- +  10.7:2-1- 20^5 +  2r)r»  +  34a;5  +  48x« 
+  54r<  +  70.?;8  ^  48.^0  +  25a;io  +  00.?:"  +  30^12. 


5.    1  _  4*.  -(-  lOa;^ 


o(X^.3  ^  25.r»  —  24rr''  +  lGa:«. 


177.    I.  {<i  +  *)S  («  +  ^)'  (^'  +  ^)  • 
4.  ^■<-  4-  2/)S  (^  +  i/)-'  C-^'  +  ^) 


h 


178.17.  '< 


(^  -  oy 


468 


AKSWA'BS. 


:\    III-. 


M 


184.    I.   10  +  3(5\/5  -  2v;J  —  3^10).     2.  37a/^  —  17. 

4.  a  +  b  -\-  c  +  d  +  2  {'s/ab  +  Vac  +  V«f/  +  's/bc 

-\  y/bd  +  '\/al). 

4       1 

12.1.     13.^/2(^/^  +  1).     17.  (2;-2/)^[(a:-y)^-l], 


19. 


Vrt  +  b 


20. 


«a;  +  ^      ^^     (rt  —  ;?;)^  +  1 
(a  +  x)^  —  1 


«:c 


21.  - 


« c^K        («'  -  30)^-        V^'-y        a/1  -  .-^2        7\/i5 

185.      I.    ^^— ^ TT-'       2.    — ^-       3.    — 4 


y         "      1  —  a;  45 

5.  -:^.     6.  5V3.     7.  -^.ly-     ^-  — r^T^- 


«  — G 

•2\/3 
3 

(V^-  +  Vy)^      .       rt'^  +  rt  {x.  +  ?y)^  -  2  (.7:  +  y) 

g. ■'     10. • 

X  —  y  a  —  X  —  y 

9^15  4-41  {y/x—  ViT^iY 


II. 


•y 


13.  — ^-^-^ '-•     14-  (^«  +  1)'  —  a 


i. 


a* 


15- 


;«  +  V^  —  d^ 


a 


187.    I.  .r^  +  2.c^  =  (:*;-  //)-2  -  y\ 

2.  x^  +  4.17/  =  {x  +  2//)3  —  hf. 

3.  .?;2  +  Grti;  =  {x  -r  ^af  —  9«2. 

4.  4:t;2  +  4.Ty  =  {^Ix  +  ?/)2  —  y". 

190.1.4     2.  ^-i^^-.     3.   («  +  ^)'.     4-6.     5.  v^. 


7'^ 


"5' 


_«v 

6.    ^<^U^^       7.    (ft2  _  J)2)mq>np^       9.    (^4  _  2«aZ,2  ^  2«<)^. 

10.     0^  -\-   n.        II. r-        12.     - 


(1  _w2)i         "'"    (1-922)2 

191.    I.  6,  12,  4.     2.  15,  12.    3.  47,  35.     4.  IG.     5-  «  +  1. 

6.   8,  16.     7.  64,  512.     8.   16,  48. 

O.     10.    lO.        10.     7— TT,,     ; rr,'        H.     ^O,   3d. 


ii 


.  y/V'Z  -  17. 

-  Vad  +  Vdc 

i^  +  y).     II.  1. 

7\/l5 

''  -T5— 

(rt  —  VxY 

„         • 

fr  —  X 

-  ^  {^  +  y) 

-y 


5-  V«*. 

G.     5.  a  +  L 
.  28,  36. 


ANSWERS. 


460 


12.    w  =: 


y 


rtc  —  ac 


\/  itb'  -  ah 


X 


=v. 


h'c  —  /yc' 
ah'  —  «^* 


13.   10,  U. 


14. 


(W     +  ?i2y^ 


J 5-  tVA 


195. 


I.   r>; 


-10,   - 


2 


y  =  ±  S.    4.  y  =  a±b. 


2.  V  = 


5.  a:  =  —  fl  or  —  5,     7.  a:  =  «(l  ±  V^). 


8. 


y 


.(- 1  ±  Vi2'.)). 


9-  y 


10.  a; 


±V! 


2 


I.    ±  21,  ±  27.     2.  4  and  10. 


3"    ±1:.     4.  'JG,  34.     5.   10,  15,30. 
6.  0,  10,  14,  18,  or  -18,  -14,  -10.  - 


7.  35.     8.  21  turkeys,  25  chicken 


G. 


9.   12.     10.   10. 


19G, 


203. 


I.  250.     12.  3.     13.   Length,  45;  hreadth,  3 

14.  ^^(V-im^  +  «2  +  rt)  and  ^(Vi^^^T"^"^  -  «). 

15.  72  or  108.  

1.81.     2.121.    3.225.    4.289.    5.  25G.     6.  V3±3V6. 


v. 


;(:±ViH>..©'.i.  ,y(„.+i)-. 


a^ 


I.  a;  = 


« 


2.  ri' 


5.  a;  ~  1.    6.  a;  =: 


8.  a:  =  5 


or 


9 


—  a.     3.  a;  =  13. 

—  1  ±  VlG(<Tq~f 

8 


4. 2: 


50. 


7-  .^  =  16. 


9.  X  =  cfi  —  b'^±  bVb^  -  a\ 


10.  .-c  —  4.     n.  a;  =  «  or 


rt 


303. 


I.    X 


or  5. 


10 


//  =  --  or  2. 


2.  a;  =  —  4  or  +  13,  «/  =  0  or  —  17. 

3-  ^  =  -3-  ±^  V^863;  y  =  ^(l  ip  V^63). 

4.  a:  =  11  or  -  7f|,  ^  =  15  or  -  17M. 

e  'Y    01       ^«     ,<    .      ,. r.lD      „..     .»  ^  ** 


5.  a:  =  —  21  or  4 


28  or  3. 


304.  I.  a:=1.37...  or  —  0.15G...;  .y=:— 4.46...  or  -6.09G. 


y 


or  —  4 :  X  = 


12 


14 

or . 


S.  X  =  2  or  6;  y  =  6  or  3. 


A^  r\ 


AysWEIiS. 


»> 


05.  I.  ^  =  ±  1  or  i: 


X 


'Zy  or 


-4f/. 


'» 


^   -    ±  .,    <'!•    ± 


2.     7/    — 


307. 


Viij' 


:i;  =  T  .,  or  ± 


'Z 


a;  =  ±5;  ?/  =  ±:>. 


Vio 


2.  .r 


=  ±8;  y  ^-  ±3. 


3-  ^ 


('")  ±  Vo) ;  y  =  ,  (5  T  Vo). 


4.  ?/  =  7  or  2 ;  ;r  =  3  or  7.     5.  a:  =  5  or  7  ;  y  =  7  or 
G.  x=z  ±  1);  ;?/  =  :f  2.     1-  x  =  ±%  -,  y  -  ±  9. 

8.  ;r  =  ±  -7=r^;  y  = 


Vc 


--=•     9-  x=2\  y=l. 


a  +  b    "  Va  +  b 

10.  X  =  a{a±b)yy  =  h  {a  ±  h).     11.  x  =  4;  y 

13.  X 


3 


12,  .T  =  ^ ;  ?/ 
0     ^ 


1 


5. 


1 

0        -  l5'  ^  -  15* 

14.  x  =  i);  y  =  lor2',z=2orl.     15,  x  =  5;  y  =  3. 
16.  Time,  6  or  7;  rate,  7  or  6.     17.  Dist.,  30  or  4G|. 

18.  X  =  ^  (V«2  +  4^2  +  V^z^T^^i^); 
-«       _ 
y  ==  vtVrt*'  +  4^  -  V«'  -  4(^i 

19-  o  (1  ±  Vs)  and  I  (3  ±  Vo). 

20.  24  and  0,  or  —1.2  and  -18.  21.  49  and  25. 
22.  G4  and  8. 

23-  >n  +  n  qp  Vni^  -^7^2  .^nd  m  +  ?i  ±  V/^^  i-  ?il 

24.  12  men  working  12  li.     25.  8  ;  10. 
;:6.  .T  =  j-G;  y  =  ±4.     27.   11;  3. 

210.   7.  14075.     8.  5050.     10.  n\     11.  71^  +  n. 

\2.  Lowest,  140  —  (jm  ;  all,  137//J  —  3^2^ 

i5.  0,  2,  4,  G,  8.     17.951.     iS.  4,  10,  IG.     19.   11  or  8. 

21.  10  or  IG  d.     22.  0  days.     23.  2,  5,  8,  11,  14. 

25.  2,  G,  10,  14,  18,  22,  26,  30,  34,  38.     27.  3,  5,  ... .  29. 


/ 


28.  a^  «  -f-  - 


-,  a  +  .-V-  -',  etc. 


M-  1 '  /  +  1 

212,   6.   Last  nail,  *2147483G.48  ;  all,  '<!42049G72.95.     7.  246. 

12.  5  or  ^. 
5 

'-514.    I    ^'    2    '2     -2     ^       i*r^r^.    m^—2?n 

I.  -    2.  2.  3.  --^.  4.  ^.  5.  ^.  G.  ^^.   7.  --s:^-^- 


io 


i'i= 


ANSWERS. 


471 


-4?/. 

a/10 
i;  ^^-  ±3. 


3r7;  y  =  7ora. 


o; 


^=  ±9. 


9-  ^"=2;  y=:l. 

j^'  =  4  ;  y  =  5. 
15' 
',  30  or  46f. 


and  25. 


19-  11  or  8. 
1,14. 

3,  5,  ... .  ^<j. 


2.95.     7.  246. 


7- 


Wi3_l" 


8. 

8. 
316.   I. 


1'2  —  C)  -\-  3  —  -  -f-  etc.,  ad  inf. 


from  A  to  li. 


3-  1-     4-  rr.'     5 


707 

iioo" 

$210.74.     3.  2.7232oa. 


10 


11 


110 


7-  ^, 


108 


000 


4.  a 


(^+4r-> 


(^-iior-r+Bj 


and 


( 


1  + 


100/ 


326  (rt). 

1.  440. 

16. 

1.     17.  • 

23. 

1 
-4-     2^ 

237.   I. 

.43  =  A 

jIjO    =     — 

2. 

ylg  =  IG 

4- 

(^+n;or' 


3.   74.     4.  148.     5.  0.     6.  0. 
4.     18.   10.    19.  20.     20.  .35.    21.56.    22.0. 

.   —  1.     3.   74.     4.  148.    5.  0.     6.  0. 

I  —  Aq-,  A3  =  —Aq  ;  A^  =  —A^ ;  etc. ; 
A,. 

Jj  -15^0-     3-  -I5  =-i.3J.  +3C.1„. 


4o  +  «/-')•     5- 


r(r'^-l) 


.1 


6.  Jg  =  kA.  +  yl 


0 ' 


228.  I.  71  —  4:  terms  are  omitted.     5.  5  _  2 


.^le  =  (120^-5+ 061-3+ 9;^-)^ 
+  (120X-*  +  30P4-  l)^4o 


I.  120.     2.  720.     3.  40320.     4.  35.     5.  56. 


I. 

230.  I. 

6. 

232.  2. 

233.  6. 

245.   I. 


247. 


ro. 


2^.3.     2.  23.32.     3.  22.5.13.     4.  1.32.     5.  32.51 

28. 

7.     3.  1.    4.  1.     5-  4. 

First  wheel  =  7,  second  =  5,  third  =  3  turns. 

113 


355 


17 

58' 


10 


ox  4-  1 


bc  +  1 


3  {ox  +  1)  4-  ic 


a 


{be  +  1)  +  c 


251.   2.    ;.    4.  720.    5.  24. 


•» 


472 

6. 

7- 

*^5:j.  I. 

254.  4. 
!^55.  I. 
251,  3. 

9- 

258.  2. 

361.  3 
363.  I. 

367.  I. 

7- 

15- 

360.  I. 

3- 

5- 
6. 

7. 


.43^6rTFA'i?^. 


(a)  21G0  even,  2880  ocUl  ;  (b)  144;  (c)  720  ;  (d)  576. 

720.     8.    120.     9.  120.     10.  120.      11.  12.     12.  72. 

144.     14.  720. 

3(J0.     5.   GO.     6.  00     7.  24. 

720.     2.   120.     3.  48.     4.  4.     5.  2880.     6.  14400. 

140.     5.  048G480. 

5.     2.  5.     3.  8.     4.  C.     5.  IG.     6.  17. 

3,  21,  36.     5.  10.     7.  3.    8.  3. 

(a)  1 ;  (5)  3 ;  (r)  6  ways.     10.  3003. 


35 


/27i  —  1\  I'Zu  —  \\ 


15  and  20.     5.  (^^)- 

120.     2.  240.     3.  2».     4.  81  routes. 


%      ^'  3'  3'    ^'  36'  18*    "^  36" 


^      8 
5*    ^'3 


2  1 
3'  3 
1  5 

'Zmn 


1 

5-6' 


9-I4- 


3  1 

10.  — •  II.  -' 

10  o 

16. 


13- 

2"' 


6. 
3 

—  • 

7 


14 


•  27' 


n 
^7-  9-„- 


(/«  +  w)(m  +  M—  1)      '"'10 
2,2  63     „    1  7.9 

5  "^^^15-     -^-"'SO'  ^'80'  ^'80'  ^'80 

3 

2  to  1  ill  favor.    A-  -r-,' 

14 

dmn  {m  —  1) 


(in  +  n)  {m  -\-  u  —  1)  (m  +  u  —  2) 

^  =  7'  ^  =  2l'  ^==2l'  ^"=7'  ^^2l*  *'="2i" 


1 

2^ 
2n 


1 


-,  X  =  -'.     8.  A,;,;  B  i. 


4 

1        2» 


3 

on 


2« 


10. 

12. 

371.    I. 
374.   I. 


2  2«  —  1 '  22  2«  —  1  '  ■  "  '  2«  2"  —  1 
36    3^    25 

91'  '91'  91' 

The  chances  are  41  to  25  in  favor  of  the  first  purse, 

^JL    ^     ^     i?_    J:        -,    .?il  ^^ 

243'  243'  243'  243'  243*     ''  2048* 

(a)  0.429  ;  (b)  0.159;  (c)  0.813;  (^Z)  0.655 

(e)  0.371;  (/)  0.110;  (^)  0.151;  {/i)  0.025. 


3-   o 


27 


ANSWERS. 


473 


ic)  r20  ;  {(l)  576. 

".  12.    12.  n. 


>80.    6.  14400. 
17. 


1      .    1 
•6*     ^-r 


3 

7" 


^^-  7-     ^4-  W 
n 

'80 


1 

'3' 

_  l__ 

~  2"  — T* 


le  first  purse. 

16 
^'  27' 


.655; 
025. 


2.  69.     4.  $296.30.     5.  0.4533. 

6.  *1000;  $1666.67;  $2111.11.     7.  «  [1  _  (I  -  ;M. 

8.  0.1123.     9.  $1894.     10.  $1224. 

378.    I.  140.     2.  70.     3.  112.     5.  22^.     6.  28(y-l). 

8.  54^2.     p, 

383.  I.  l  +  a;4-ar2H-ar3  +  etc.     2.  1  +  2a; +  22.i-8+ 232^5  + etc. 

3.  1— 2a:4-2.T2_3;e3  +  etc.     4.  l  +  2:<;  +  2.t;2+ 2.1-3+ etc. 
5-  1— a:— .^•'  +  5a:3— 7:^4— etc.  6.  l+a;+a:H2^+ari  +  etc. 
7-  1  —  4.2;  +  8.1-2  —  4.1-3  _  iQ^i  ^  ^^(3^ 

8.   1  —  2a;  +  2a;2  --  x^  —  o^ -\-  etc. 
383.  1.  1— 3a:  +  3.?;2— 3.T3  +  etc.    2.  l  +  2a;  +  a;2_a^_2a:4_etc. 
388.  1.  ^ll?^I±i).    2    ^^  (^^  +  1)  -  .9  (..  +  1) 


2 


2 


y?(M  4-l)--w(^/;-l)           ;7(io-f  2^«-l) 
3.  3    —  4. 

6.  3?i2  _  3;^  4.  1. 

389.  I.  165.   2.  ?il!y:iitLt^-i:A(^JM+  2) 

1.2-3  ~~* 

7^(/?  +  l)(7^  +  2)-0/?-  1)  wi  (m  +  1) 
^'  I.2T3 

393.  I.  S^  =  210;  ^Vg  =  2870;  .S'3  =  42665. 

r(4/-2  — 1) 
3 


2.  S^  =  r2;  ,S' 


3.  ^,  =  r(;.+  1);  ^,  =  ?^iL±Jil!rJli). 

o 

4.  iV^a  =3;9<7-3(/>  +  ^)  +  5; 

JVp  z=  spq  —  IAIzlI)  (3^^  +  3r^  -  25  +  1). 

5.  5fl  +  156  +  55c.     6.  ^L  +  ^L+i(a^^^  +  lj' 

2  lA L_\ 

w  +  3        •  2\3       2;«  +  3/ 
.    2/111 1_  1  1     \ 

^' 312"^ 3  "^4    «  +  2~;r+T"';rfi/* 


395.  I.  ^  -      ^ 


^'  20+2~^Tl~^TT2)" 


5. 


1 
a 


474 


AA\SWERS. 


2im.  I. 


2^(1- /«-')   ,   rt  [1  -  {'in  -  1)  r"l       T  rt  (1  4-  r) 


+ 


l-r 


and 


(1  -  yf 


Umr 


and  -, 


'4a 


{i-rf 


hr  (I  —  ?'")       (tr--{a  +  ^^J)  r»+i  (J  4.  «);•—«?•' 


(1  -  yf 


+ 


1  — r 


and 


(1 


_.  r'ka 


39 


300.  2.  A5  =  -  305 ;  A j  =  „  /'^  -  "^  i2 .._  2:  ..y  5 


:e 


341°  5'  10".9+(;;.  -  1)  (T  0'  0".0)  -  (7i_l)(7e_2)' 


4.  495  +  15  (m  —  5) 


0 


(m  —  0)  {n  —  G) 


2 


Morning  of  May  23  or  Apr.  24. 


304.  I.  1.    2. 


a  in  1 

,•     3.  — •     4.  -• 


5.  '2a.     G.  —  1. 


308.  I.  VS  =  2.828437  ;  V2  =  1.414214. 

1  .  _  1-J.   3  _  ia-J3   3  _  1.1.3-5 

JJ"^'       ^  4'^'"       "2.4-0''^        2-4. G.8' 


2.  1  —  ^.r  —  --4.^'"  —  ~^T—.x^  —  --^^-^Zx^  —  etc. 


3.  Goncial  term  ~  — 


1-1. 3. 5-7.... 2/- 3    . 


2-4-G.8 2i 

,      ,,,1.3.5....(2/— 1)    .  /m\l 

4-(-^)^-2:4:g:^-:T27-'^-*-  5.  {j)~, 

(—l)(ni  —  1){2»}  —l)....[{i  —l)m—  1] 


6. 


il  m^ 


\-m      a-  m)  (1  -  2^;0 
7.  1  +  1  +  — ^T-  + 31 

(1  —  m)  (1  —  2;m)  (1  -  3m) 


+ 


4! 


+  etc. 


W^lb'^  l-'Zb^'^""'^     1-2.3. ...i    h^V 


w  (y;^  -f  1)  {m  +  2)  \ 

''"         1.2.3a;'«"3        +••••/■ 


I 

I 


•  and  'liLt^X 


■  +  5. 

.  {n-l){n-2)". 


-  1. 


—  etc. 

f-3 

x^. 

|1. 

.T^ 

h^i  — 

■  1 

llIiNTS  ON  A  COURSE  IN  AI)\  ANCKl)  ALlll-KRA. 


3m) 


+  etc. 


2) 


+ 


...). 


For  tlio  iM'ncfit  of  students  who  may  contemplate  a  course  of  rcadinjj: 
in  tliL-  vtu'ious  hranches  of  Advaiired  Al^fl)ra,  tlu^  followiuir  list  of  niilt- 
j(  cfs  and  Ijookf  has  been  iJrc'iiarL'd.  As  a  {general  ruK-,  tlic  most  extended 
and  tliiiroufi:li  treatises  are  in  the  (lernian  Laii^nui^e,  wlulo  the  French 
works  are  noted  for  elegance  and  simplicity  in  treatment. 

To  pursue  any  of  tlie^e  'uibjects  to  advantage,  the  student  ehouhl  be 
familiar  with  the  Ditlerential  Calculus. 

I.  THE    GENERAL   THEORY  OF    EQUATIONS.— In   English.   ToD- 

liUXTEU's  is  the  work  most  read. 
Berukv,  A/;/rbre  Snp'rienir,  2  vols.,  8vo,  is  the  standard  French  wcjrk, 

covering  all  the  collateral  subjects. 
JoiiD.VN,  IVu'orie  dcs  SubntUntiDiu  et  des  EquationH  Ahjrhriqucs,  1  vol.,4to. 

is  the  largest  and  most  exhaustive  treatise,  but  is  too  abstruse  for 

any  but  experts. 

II.  DETERMINANTS— Bat.tzei^,   Theorio.  drr  Detrrmivrivtcn,   is  the 

standard  treatise.  Tliere  is  a  French  but  no  English  translation. 
A  recent  I'higlish  work  is  Koiu-:ht  F.  Scott.  T/ic  Tluonj  of  Bitci'- 
minants  and  their  AppUattions  in  Aiudynis  (ind  Geometry. 

III.  THE  MODERN  IlKJHER  ALCJEHRA,  resting  on  the  theories  of 

Invariants  and  Covariants. 
Salmon,    Lessons  introductory  to  the  Modern   Higher  Atgehra,  is  the 

standard  English  work,  and  is  especially  atlapted  for  instruction. 
Clebscii,  Theorir.  der  binaren  Algibniischtii  Formm,  is  more  exhaustive. 

in  its  sp(!cial  branch  and  reijuires  more  lamiliarity  with  advanced 

systems  of  notation. 

IV.  THE  THEORY  OF  NUMBERS.      There   is  no   recent   treatise   in 

English.  Gauss,  Disquidtioneti  Arithmetiem,  and  I.kgknduk, 
Theorie  dcs  Nombres,  are  the  old  standards,  but  the  latter  is  rare 
and  costly.  Lejeumc  Diuiciii.rvr,  Vorlemnigen  iiber  Zahlciitheorie, 
is  a  good  (lerman  Work.  There  is  also  a  chapter  on  the  subject  in 
Se«ket,  Algebre  Superienre.  . 

V.  SERIES.— This  subject  belongs  for  the  most  part  to  the  Calculus,  but 

Catalan,  Trnite  elemeiitaire  des  SHrPs,  is  a  very  convenient  little 
French  work  on  those  Series  which  can  be  treated  by  Elen  eutary 
Algebra. 

VI.  QUATERNIONS.— Tait,  Elementary   Trexdisc  on  Qnnternions,  is 

pr(>i)ared  especially  tor  students,  and  contains  many  exercises.  The 
original  works  of  Hamilton,  Lectures  on  Qjtedcrnions  and  Elements 
of  Qnnternims,  are  more  extended,  and  the  latter  will  be  found 
valuable  for  both  reading  and  refereuce. 


